Section 1 Number Theory

**Chapter 1**

**Abstract**

**1. Introduction**

**3**

Transform

*Rajesh Cherian Roy*

and inverse transforms are presented.

A New Integer-to-Integer

This chapter presents a detailed analysis of an integer-to-integer transform that

is closely related to the discrete Fourier transform, but that offers insights into signal structure that the DFT does not. The transform is analyzed for its underlying properties using concepts from number theory. Theorems are given along with proofs to help establish the salient features of the transform. Two kinds of redundancy exist in the transform. It is shown how redundancy implicit in the transform can be eliminated to obtain a simple form. Closed-form formulas for the forward

**Keywords:** transforms, discrete transforms, integer transforms, DFT, MRT

Transforms are tools used in signal processing to arrive at deeper insights into the underlying structure of signals. The discrete Fourier transform (DFT), discrete cosine transform (DCT) [1], discrete sine transform (DST), discrete Hartley transform (DHT) [2], and discrete wavelet transform (DWT) are significant discrete transforms. Discrete transforms are characterized by their basis matrices. The Haar transform is distinct in that its basis matrix has only 1, 1, or 0 as elements. The Walsh-Hadamard transform basis matrix is entirely composed of 1 and 1. The discrete Fourier preprocessing transform [3] also has only 1, 1, or 0 in its basis matrix. A new discrete transform, M-dimensional real transform (MRT) based on linear congruences was first proposed [4] for two-dimensional signals. Its basis matrix contains 1, 1, and 0 only. However, it has a high level of redundancy. The orthogonal discrete periodic radon transform (ODPRT) [5] is another transform that is based on linear congruences. MRT has been applied to image compression by making use of a set of unique MRT coefficients [6, 7]. A one-dimensional form of the MRT exists [8]. The Haar transform is related to the Walsh-Hadamard transform [9]. The MRT is related [10] to the Haar transform. One can be obtained from the other through a sequence of bit reversal operations on the rows and columns. Similarly, the Hadamard transform and the MRT can be derived from each other [11]. In [12], 2-D gcd-delta functions that contain only zeroes and ones are used to generate integer 2D DFT pairs. The Weyl transform which has binary valued inner elements is described in [13]. Integer-to-integer approximations of DST are still an active research area, as can be seen in [14]. In comparison to most integer-tointeger transforms, the MRT is distinguished by the utmost simplicity of its kernel. In this context, the MRT can be grouped together with Haar and Walsh-Hadamard transforms as regards the contents of their basis matrices. This chapter presents a

#### **Chapter 1**

## A New Integer-to-Integer Transform

*Rajesh Cherian Roy*

#### **Abstract**

This chapter presents a detailed analysis of an integer-to-integer transform that is closely related to the discrete Fourier transform, but that offers insights into signal structure that the DFT does not. The transform is analyzed for its underlying properties using concepts from number theory. Theorems are given along with proofs to help establish the salient features of the transform. Two kinds of redundancy exist in the transform. It is shown how redundancy implicit in the transform can be eliminated to obtain a simple form. Closed-form formulas for the forward and inverse transforms are presented.

**Keywords:** transforms, discrete transforms, integer transforms, DFT, MRT

#### **1. Introduction**

Transforms are tools used in signal processing to arrive at deeper insights into the underlying structure of signals. The discrete Fourier transform (DFT), discrete cosine transform (DCT) [1], discrete sine transform (DST), discrete Hartley transform (DHT) [2], and discrete wavelet transform (DWT) are significant discrete transforms. Discrete transforms are characterized by their basis matrices. The Haar transform is distinct in that its basis matrix has only 1, 1, or 0 as elements. The Walsh-Hadamard transform basis matrix is entirely composed of 1 and 1. The discrete Fourier preprocessing transform [3] also has only 1, 1, or 0 in its basis matrix. A new discrete transform, M-dimensional real transform (MRT) based on linear congruences was first proposed [4] for two-dimensional signals. Its basis matrix contains 1, 1, and 0 only. However, it has a high level of redundancy. The orthogonal discrete periodic radon transform (ODPRT) [5] is another transform that is based on linear congruences. MRT has been applied to image compression by making use of a set of unique MRT coefficients [6, 7]. A one-dimensional form of the MRT exists [8]. The Haar transform is related to the Walsh-Hadamard transform [9]. The MRT is related [10] to the Haar transform. One can be obtained from the other through a sequence of bit reversal operations on the rows and columns. Similarly, the Hadamard transform and the MRT can be derived from each other [11]. In [12], 2-D gcd-delta functions that contain only zeroes and ones are used to generate integer 2D DFT pairs. The Weyl transform which has binary valued inner elements is described in [13]. Integer-to-integer approximations of DST are still an active research area, as can be seen in [14]. In comparison to most integer-tointeger transforms, the MRT is distinguished by the utmost simplicity of its kernel. In this context, the MRT can be grouped together with Haar and Walsh-Hadamard transforms as regards the contents of their basis matrices. This chapter presents a

detailed study of the MRT in its one-dimensional form and offers an analytical path that justifies placing the MRT beside the Haar transform in the family of discrete transforms.

ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* þ *M* (5)

<sup>4</sup> and data elements *x*1, *x*3, *x*5, *x*<sup>7</sup> form the

ð Þ ð Þ *nak <sup>N</sup>* ¼ *p* (6)

*<sup>N</sup>* ¼ *p* þ *M* (7)

*<sup>b</sup>* (10)

*<sup>a</sup>* ¼ *nb* (11)

ð Þ ð Þ *nbk <sup>N</sup>* ¼ *p* þ *M* (8)

*k*

*<sup>k</sup>* can be

*<sup>k</sup>* can

*<sup>k</sup>* be

(9)

The group of data elements whose indices satisfy the congruence relation ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* is defined as the positive data group or positive set of the 1-D MRT

relation ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* þ *M* is defined as the negative data group or negative set of the

An MRT coefficient has two indices, frequency and phase. By formal definition

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup> k*

**Proof**. The elements *na* that are in the positive group of the MRT coefficient *Y*ð Þ *<sup>p</sup>*

*<sup>a</sup>* that are in the negative group of the MRT coefficient *<sup>Y</sup>*ð Þ *<sup>p</sup>*

*<sup>b</sup>* that are in the negative group of the MRT coefficient *<sup>Y</sup>*ð Þ *<sup>p</sup>*þ*<sup>M</sup>*

*<sup>N</sup>* ¼ *p* þ *M* þ *M* ¼ *p* þ *N*, which *can be* written *as*

*na* ¼ *n*<sup>0</sup>

*n*0

<sup>4</sup> .

of the MRT, phase has values in the range 0, ½ � *M* � 1 . Although by definition, *p*∈ ½ � 0, *M* � 1 , number theory allows *p* ∈½ � 0, *N* � 1 . *p* is referred to as a valid phase for a given value of *k* if *k*∣*p*. For example, for *N* ¼ 6, if *k* ¼ 2, then *p* ¼ 0, 2, 4 satisfy *k*∣*p*, and hence, these are valid phases for this value of *k*. A phase *p* is defined to be

*Y*ð Þ *<sup>p</sup>*

*n*0 *ak*

The elements *nb* that are in the positive group of the MRT coefficient *Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup>*

*<sup>k</sup>* The group of data elements whose indices satisfy the congruence

*<sup>k</sup>* . For example, the data elements *x*0, *x*2, *x*4, *x*<sup>6</sup> form the

coefficient *Y*ð Þ *<sup>p</sup>*

1-D MRT coefficient *Y*ð Þ *<sup>p</sup>*

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

positive set of the MRT coefficient *Y*ð Þ <sup>0</sup>

negative set of the MRT coefficient *Y*ð Þ <sup>0</sup>

an allowable phase index if *p* < *M*.

**2.3 Theorem 1: periodicity**

can be found as the solutions of

The elements *n*<sup>0</sup>

found as the solutions of

be found as the solutions of

The elements *n*<sup>0</sup>

**5**

found as the solutions of

*n*0 *bk*

*n*0 *bk*

*<sup>N</sup>* ¼ *p*

From Eqs. (6) and (9), it can be inferred that

From Eqs. (7) and (8), it can be inferred that

The MRT is defined and its salient characteristics are presented by way of theorems and their proofs. The redundancy inherent in the MRT is studied and the redundancy-removed version of the MRT named the unique MRT (UMRT) is presented. The inverse UMRT is presented. Described next is a form of the UMRT basis matrix which can be indexed using only 2 indices.

#### **2. Mapped real transform (MRT)**

#### **2.1 Forward 1-D MRT**

The MRT *Y*ð Þ *<sup>p</sup> <sup>k</sup>* of a 1-*D* sequence *xn*, 0≤*n*≤ *N* � 1 is defined [7] as

$$\begin{aligned} Y\_k^{(p)} &= \sum\_{\forall n \Rightarrow ((nk))\_N = p} \mathbf{x}\_n - \sum\_{\forall n \Rightarrow ((nk))\_N = p + M} \mathbf{x}\_n \\ k &= \mathbf{0}, \mathbf{1}, 2...N-1, \ p = \mathbf{0}, \mathbf{1}, 2...M-1, \text{and} \ M = \frac{N}{2} \end{aligned} \tag{1}$$

In Eq. (1), *k* can be considered analogous to the frequency in DFT, and *p* signifies the phase. Thus, 1-D MRT produces *M* arrays each of size *N*, given a signal of size *N*. Hence, *MN* coefficients have to be computed using only real additions. Another expression for the 1-D MRT [7] is

$$Y\_k^{(p)} = \sum\_{n=0}^{N-1} A\_{k,p,n} x\_n, \; 0 \le k \le N-1, \; 0 \le p \le M-1 \tag{2}$$

$$\begin{aligned} \mathbf{1} \, \left. \dot{\mathbf{f}} [nk]\_{N} = p \\ A\_{k, p, n} = -\mathbf{1} \, \dot{\mathbf{f}} \, \|nk\|\_{N} = p + M \\ \mathbf{0} \, otherwise \end{aligned} \tag{3}$$

Thus, the kernel *Ak*,*p*,*<sup>n</sup>* maps the data *xn* into the 1-D MRT *Y*ð Þ *<sup>p</sup> <sup>k</sup>* . For example, Let *x* ¼ ½ � 95 23 61 49 89 76 46 2 , *N* ¼ 8. Then, *Y*ð Þ *<sup>p</sup> <sup>k</sup>* , the corresponding MRT of *x*, is

$$\begin{aligned} Y\_k^{(0)} &= \begin{bmatrix} 441 & 6 & 77 & 6 & 141 & 6 & 77 & 6 \end{bmatrix} \\ Y\_k^{(1)} &= \begin{bmatrix} 0 & -53 & 0 & 47 & 0 & 53 & 0 & -47 \end{bmatrix} \\ Y\_k^{(2)} &= \begin{bmatrix} 0 & 15 & 48 & -15 & 0 & 15 & -48 & -15 \end{bmatrix} \\ Y\_k^{(3)} &= \begin{bmatrix} 0 & 47 & 0 & -53 & 0 & -47 & 0 & 53 \end{bmatrix} \end{aligned}$$

#### **2.2 Composition of MRT coefficients**

One set of elements that combine to form an MRT coefficient contains those elements satisfying the congruence ð Þ *nk <sup>N</sup>* ¼ *p*, and another set contains those satisfying the congruence ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* þ *M*. Thus, the two relevant congruences are:

$$((nk))\_N = p \tag{4}$$

*A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

detailed study of the MRT in its one-dimensional form and offers an analytical path that justifies placing the MRT beside the Haar transform in the family of discrete

The MRT is defined and its salient characteristics are presented by way of theorems and their proofs. The redundancy inherent in the MRT is studied and the redundancy-removed version of the MRT named the unique MRT (UMRT) is presented. The inverse UMRT is presented. Described next is a form of the UMRT

*<sup>k</sup>* of a 1-*D* sequence *xn*, 0≤*n*≤ *N* � 1 is defined [7] as

∀*n*)ð Þ ð Þ *nk <sup>N</sup>*¼*p*þ*M*

*xn*

*Ak*,*p*,*nxn*, 0 ≤*k*≤ *N* � 1, 0≤*p* ≤ *M* � 1 (2)

ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* (4)

2

(1)

(3)

*<sup>k</sup>* . For example,

*xn* � <sup>X</sup>

*<sup>k</sup>* <sup>¼</sup> 0, 1, 2…*<sup>N</sup>* � 1, *<sup>p</sup>* <sup>¼</sup> 0, 1, 2…*<sup>M</sup>* � 1, and*<sup>M</sup>* <sup>¼</sup> *<sup>N</sup>*

1 *if nk* ½ � *<sup>N</sup>* ¼ *p Ak*,*p*,*<sup>n</sup>* ¼ �1 *if* ∥*nk*∥*<sup>N</sup>* ¼ *p* þ *M* 0*otherwise*

*<sup>k</sup>* ¼ ½ � 441 6 77 6 141 6 77 6

*<sup>k</sup>* ¼ ½ � 0 47 0 � 53 0 � 47 0 53

One set of elements that combine to form an MRT coefficient contains those elements satisfying the congruence ð Þ *nk <sup>N</sup>* ¼ *p*, and another set contains those satisfying the congruence ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* þ *M*. Thus, the two relevant congruences are:

*<sup>k</sup>* ¼ ½ � 0 � 53 0 47 0 53 0 � 47

*<sup>k</sup>* ¼ ½ � 0 15 48 � 15 0 15 � 48 � 15

Thus, the kernel *Ak*,*p*,*<sup>n</sup>* maps the data *xn* into the 1-D MRT *Y*ð Þ *<sup>p</sup>*

In Eq. (1), *k* can be considered analogous to the frequency in DFT, and *p* signifies the phase. Thus, 1-D MRT produces *M* arrays each of size *N*, given a signal of size *N*. Hence, *MN* coefficients have to be computed using only real additions.

basis matrix which can be indexed using only 2 indices.

**2. Mapped real transform (MRT)**

*Number Theory and Its Applications*

*Y*ð Þ *<sup>p</sup>*

Another expression for the 1-D MRT [7] is

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* <sup>¼</sup> <sup>X</sup> *N*�1

*n*¼0

Let *x* ¼ ½ � 95 23 61 49 89 76 46 2 , *N* ¼ 8.

*Y*ð Þ <sup>0</sup>

*Y*ð Þ<sup>1</sup>

*Y*ð Þ<sup>2</sup>

*Y*ð Þ<sup>3</sup>

**2.2 Composition of MRT coefficients**

*<sup>k</sup>* , the corresponding MRT of *x*, is

*<sup>k</sup>* <sup>¼</sup> <sup>X</sup>

∀*n*)ð Þ ð Þ *nk <sup>N</sup>*¼*p*

**2.1 Forward 1-D MRT**

The MRT *Y*ð Þ *<sup>p</sup>*

Then, *Y*ð Þ *<sup>p</sup>*

**4**

transforms.

$$((nk))\_N = p + M \tag{5}$$

The group of data elements whose indices satisfy the congruence relation ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* is defined as the positive data group or positive set of the 1-D MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* The group of data elements whose indices satisfy the congruence relation ð Þ ð Þ *nk <sup>N</sup>* ¼ *p* þ *M* is defined as the negative data group or negative set of the 1-D MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* . For example, the data elements *x*0, *x*2, *x*4, *x*<sup>6</sup> form the positive set of the MRT coefficient *Y*ð Þ <sup>0</sup> <sup>4</sup> and data elements *x*1, *x*3, *x*5, *x*<sup>7</sup> form the negative set of the MRT coefficient *Y*ð Þ <sup>0</sup> <sup>4</sup> .

An MRT coefficient has two indices, frequency and phase. By formal definition of the MRT, phase has values in the range 0, ½ � *M* � 1 . Although by definition, *p*∈ ½ � 0, *M* � 1 , number theory allows *p* ∈½ � 0, *N* � 1 . *p* is referred to as a valid phase for a given value of *k* if *k*∣*p*. For example, for *N* ¼ 6, if *k* ¼ 2, then *p* ¼ 0, 2, 4 satisfy *k*∣*p*, and hence, these are valid phases for this value of *k*. A phase *p* is defined to be an allowable phase index if *p* < *M*.

#### **2.3 Theorem 1: periodicity**

$$Y\_k^{(p)} = -Y\_k^{(p+\mathcal{M})}$$

**Proof**. The elements *na* that are in the positive group of the MRT coefficient *Y*ð Þ *<sup>p</sup> k* can be found as the solutions of

$$((n\_ak))\_N = p \tag{6}$$

The elements *n*<sup>0</sup> *<sup>a</sup>* that are in the negative group of the MRT coefficient *<sup>Y</sup>*ð Þ *<sup>p</sup> <sup>k</sup>* can be found as the solutions of

$$(\left(n\_a'k\right))\_N = p + \mathcal{M} \tag{7}$$

The elements *nb* that are in the positive group of the MRT coefficient *Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup> <sup>k</sup>* can be found as the solutions of

$$((n\_b k))\_N = p + M \tag{8}$$

The elements *n*<sup>0</sup> *<sup>b</sup>* that are in the negative group of the MRT coefficient *<sup>Y</sup>*ð Þ *<sup>p</sup>*þ*<sup>M</sup> <sup>k</sup>* be found as the solutions of

$$\begin{aligned} \left( \left( n\_b' k \right) \right)\_N &= p + M + M = p + N, \text{which can be written as} \\ \left( \left( n\_b' k \right) \right)\_N &= p \end{aligned} \tag{9}$$

From Eqs. (6) and (9), it can be inferred that

$$n\_a = n'\_b$$

From Eqs. (7) and (8), it can be inferred that

$$n'\_a = n\_b$$

$$n'\_a = n\_b \tag{11}$$

From Eqs. (10) and (11) and the definition of MRT in Eq. (1),

$$\begin{aligned} Y\_k^{(p)} &= \sum n\_a - \sum n\_b\\ \sum n\_b' - \sum n\_a' \\ -\left(\sum n\_a' - \sum n\_b'\right) \\ -Y\_k^{(p+M)} \\ \therefore Y\_k^{(p)} &= -Y\_k^{(p+M)} \end{aligned}$$

phases and hence *<sup>M</sup>*

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

phase index *<sup>p</sup>* � *<sup>M</sup>*. However, the relation *<sup>Y</sup>*ð Þ *<sup>p</sup>*

phase be *p*2. The valid phase corresponding to *p*<sup>2</sup> is *p*<sup>0</sup>

smallest integer such that *qk*> *M*. Then,

phase. There is no valid phase *p*<sup>0</sup>

The distance between *p*<sup>0</sup>

between *p*<sup>1</sup> and *p*<sup>2</sup> has to be lesser than *k*. Hence,

�*Y*ð Þ *<sup>p</sup>*þ*ck*

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*�*<sup>M</sup>*

valid phases *p* > *M*.

*Y*ð Þ *<sup>p</sup>*

*p*0

**7**

divide *M*, *<sup>t</sup>*

Hence,

*<sup>k</sup>* MRT coefficients, the phases are *p* ¼ 0, *k*, 2*k*, …, *M* � *k*.

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*�*<sup>M</sup>*

*k* ¼

*<sup>k</sup>* is still satisfied. Thus, for

<sup>2</sup> ¼ *p*<sup>2</sup> þ *M*. Let *q* be the

*<sup>k</sup>* <sup>¼</sup> *<sup>t</sup>* 2

<sup>2</sup> ∣*M*.

. Since *k* does not

*t* .

<sup>1</sup> ¼ *p*<sup>1</sup> þ *M*. The next valid phase is hence given by

These MRT coefficients have both positive and negative sets simultaneously.

When *k* is not a divisor of *M*, for a valid phase, only one among the positive or negative sets exists. The valid phases are an arithmetic series *p* ¼ 0, *k*, 2*k*, …, *N* � *k*. In this case, there is no integer *<sup>c</sup>* that satisfies *<sup>p</sup>* <sup>þ</sup> *ck* <sup>¼</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>M</sup>*, and hence, *<sup>Y</sup>*ð Þ *<sup>p</sup>*

*<sup>k</sup>* cannot be true. Thus, for *p* > *M*, where *p* is a valid phase, there is no valid

*<sup>k</sup>* to express the MRT corresponding to a valid phase index *p* > *M* in

*p*> *M*, there is an allowable but nonvalid phase *p* � *M*. We can use the relation

terms of an allowable phase index. MRT coefficients formed by valid phases have only positive sets. Thus, the positive set of the MRT coefficient with *p*> *M* is the negative set of the MRT coefficient with allowable nonvalid phase *p* � *M*. Hence, a subset of the *M* allowable phase indices are valid phases, while the other subset is made up of allowable nonvalid phases of the form *p* � *M* that are obtained from

The gap between two successive valid phases is *k*. Let there be an allowable and

<sup>2</sup> ¼ *p*<sup>1</sup> þ *qk*

*p*<sup>2</sup> is the closest allowable nonvalid phase to *p*1. *p*<sup>1</sup> is the lowest allowable valid

∴*p*<sup>2</sup> � *p*<sup>1</sup> ¼ *qk* � *M*

Since the division of any even number by an odd number produces an even

<sup>2</sup> is not an integer. For this to be true, *<sup>t</sup>* has to be odd. We know *<sup>k</sup>* <sup>¼</sup> *<sup>N</sup>*

<sup>2</sup> , *<sup>d</sup>* being an integer

2

2

<sup>2</sup> and *M* has to be lesser than *k*. Thus, the distance

*<sup>p</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>1</sup> <sup>¼</sup> *qk* � *dk*

<sup>¼</sup> ð Þ <sup>2</sup>*<sup>q</sup>* � *<sup>d</sup> <sup>k</sup>*

*p*<sup>2</sup> � *p*<sup>1</sup> is greater than or equal to *k:*

<sup>¼</sup> <sup>2</sup>*qk* <sup>2</sup> � *dk* 2

The value of 2ð Þ *q* � *d* cannot be larger than 1, since, in that case,

numerically smallest valid phase *p*<sup>1</sup> < *M*, and let the nearest allowable nonvalid

*p*0

<sup>2</sup> ¼ *p*<sup>1</sup> þ *qk*, since *q* is the smallest integer such that *qk*> *M*

Since *<sup>k</sup>*∣*N*, there is an integer *<sup>t</sup>* satisfying *<sup>N</sup>* <sup>¼</sup> *tk*. Hence, *<sup>M</sup>*

number, *k* has to be even. Since *k*∣*N* and *k* is even, it follows that *<sup>k</sup>*

*<sup>M</sup>* <sup>¼</sup> *dk*

#### **2.4 Theorem 2: existence conditions**

An MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* exists for data of order *N* if either of the following two conditions is satisfied:

Condition 1: *g k*ð Þ , *N* ∣*p*

Condition 2: *g k*ð Þ , *N* ∣*p* þ *M*

If Condition 1 is satisfied, the positive data set is not a null set. If Condition 2 is satisfied, the negative data set of the MRT coefficient is not a null set.

**Proof**. An MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* exists given *N* if congruences (4) and/or (5) have solutions. The necessary and sufficient condition for (4) to have solutions is that *g k*ð Þ , *N* ∣*p*. Similarly, *g k*ð Þ , *N* ∣*p* þ *M* becomes the necessary and sufficient condition for (5) to have solutions.

A linear congruence, ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*, if solvable, has *g k*ð Þ , *N* solutions mod *N*. Hence, there exist *g k*ð Þ , *N* solutions for *n* in the range 0, ½ � *N* � 1 , and thus, there exist *g k*ð Þ , *N* elements in the positive set. Also, if there is a member *n*<sup>0</sup> of the positive set (particular solution), the other solutions are *<sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup> <sup>þ</sup> *<sup>N</sup> g k*ð Þ , *<sup>N</sup> <sup>t</sup>*, 0 <sup>≤</sup>*t*<sup>&</sup>lt; *g k*ð Þ , *<sup>N</sup>* .

These are the other members of the positive set.

The indices of the elements in the positive (or negative) set of an MRT coefficient form an arithmetic progression,

$$m\_0 + j\mathbf{g}\_k, j = [0, \mathbf{g}(k, N) - \mathbf{1}] \tag{12}$$

where *<sup>n</sup>*<sup>0</sup> is the smallest member of the positive (or negative) set, and *gk* <sup>¼</sup> *<sup>N</sup> g k*ð Þ , *<sup>N</sup>* .

#### **2.5 Dependence of phase index on frequency**

The existence of 1-D MRT coefficients can be studied for their dependence [15] on the frequency index *k*.

If *k* and *N* are relatively prime, then *g k*ð Þ¼ , *N* 1. When *g k*ð Þ¼ , *N* 1, the MRT coefficients corresponding to *k* are in existence for all values of *p* ∈½ � 0, *M* � 1 . When *k* divides *N*, *g k*ð Þ¼ , *N k*. The condition according to Eq. (4) now becomes *k*∣*p*. There are *<sup>N</sup> <sup>k</sup>* such values of *p* in 0, ½ � *N* � 1 . These are *p* ¼ 0, *k*, 2*k*, …, *N* � *k*. The condition corresponding to Eq. (5) is now *k*∣*p* þ *M*. When *k*∣*p*, the condition *k*∣*p* þ *M* has solutions only if *k* divides *M*. Thus, an MRT coefficient has both positive and negative sets only if *k* divides *M*. Otherwise, only one among the positive or negative sets exists, for a given value of *p*.

When *k* divides *M*, the valid phases for which MRT coefficients have positive sets are *<sup>p</sup>* <sup>¼</sup> 0, *<sup>k</sup>*, 2*k*, …, *<sup>N</sup>* � *<sup>k</sup>*. There are *<sup>N</sup> <sup>k</sup>* such phases. When *k*∣*M* is satisfied, negative sets also exist for all these MRT coefficients. There are thus *<sup>M</sup> <sup>k</sup>* allowable

#### *A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

From Eqs. (10) and (11) and the definition of MRT in Eq. (1),

*<sup>k</sup>* <sup>¼</sup> <sup>X</sup>*na* �<sup>X</sup>

*<sup>b</sup>* �X*n*<sup>0</sup>

� �

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup> k*

If Condition 1 is satisfied, the positive data set is not a null set. If Condition 2 is

have solutions. The necessary and sufficient condition for (4) to have solutions is that *g k*ð Þ , *N* ∣*p*. Similarly, *g k*ð Þ , *N* ∣*p* þ *M* becomes the necessary and sufficient condi-

A linear congruence, ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*, if solvable, has *g k*ð Þ , *N* solutions mod *N*. Hence, there exist *g k*ð Þ , *N* solutions for *n* in the range 0, ½ � *N* � 1 , and thus, there exist *g k*ð Þ , *N* elements in the positive set. Also, if there is a member *n*<sup>0</sup> of the positive

The indices of the elements in the positive (or negative) set of an MRT coeffi-

where *<sup>n</sup>*<sup>0</sup> is the smallest member of the positive (or negative) set, and *gk* <sup>¼</sup> *<sup>N</sup>*

The existence of 1-D MRT coefficients can be studied for their dependence [15]

*<sup>k</sup>* such values of *p* in 0, ½ � *N* � 1 . These are *p* ¼ 0, *k*, 2*k*, …, *N* � *k*. The condition corresponding to Eq. (5) is now *k*∣*p* þ *M*. When *k*∣*p*, the condition *k*∣*p* þ *M* has solutions only if *k* divides *M*. Thus, an MRT coefficient has both positive and negative sets only if *k* divides *M*. Otherwise, only one among the positive or

If *k* and *N* are relatively prime, then *g k*ð Þ¼ , *N* 1. When *g k*ð Þ¼ , *N* 1, the MRT coefficients corresponding to *k* are in existence for all values of *p* ∈½ � 0, *M* � 1 . When *k* divides *N*, *g k*ð Þ¼ , *N k*. The condition according to Eq. (4) now becomes *k*∣*p*.

When *k* divides *M*, the valid phases for which MRT coefficients have positive

negative sets also exist for all these MRT coefficients. There are thus *<sup>M</sup>*

*a*

*b*

*<sup>a</sup>* �X*n*<sup>0</sup>

*nb*

*<sup>k</sup>* exists for data of order *N* if either of the following two

*<sup>k</sup>* exists given *N* if congruences (4) and/or (5)

*n*<sup>0</sup> þ *jgk*, *j* ¼ ½ � 0, *g k*ð Þ� , *N* 1 (12)

*<sup>k</sup>* such phases. When *k*∣*M* is satisfied,

*g k*ð Þ , *<sup>N</sup> <sup>t</sup>*, 0 <sup>≤</sup>*t*<sup>&</sup>lt; *g k*ð Þ , *<sup>N</sup>* .

*g k*ð Þ , *<sup>N</sup>* .

*<sup>k</sup>* allowable

*Y*ð Þ *<sup>p</sup>*

X

**2.4 Theorem 2: existence conditions**

**Proof**. An MRT coefficient *Y*ð Þ *<sup>p</sup>*

cient form an arithmetic progression,

on the frequency index *k*.

There are *<sup>N</sup>*

**6**

An MRT coefficient *Y*ð Þ *<sup>p</sup>*

*Number Theory and Its Applications*

tion for (5) to have solutions.

conditions is satisfied: Condition 1: *g k*ð Þ , *N* ∣*p* Condition 2: *g k*ð Þ , *N* ∣*p* þ *M* *n*0

� <sup>X</sup>*n*<sup>0</sup>

�*Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup> k* ∴*Y*ð Þ *<sup>p</sup>*

satisfied, the negative data set of the MRT coefficient is not a null set.

set (particular solution), the other solutions are *<sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup> <sup>þ</sup> *<sup>N</sup>*

These are the other members of the positive set.

**2.5 Dependence of phase index on frequency**

negative sets exists, for a given value of *p*.

sets are *<sup>p</sup>* <sup>¼</sup> 0, *<sup>k</sup>*, 2*k*, …, *<sup>N</sup>* � *<sup>k</sup>*. There are *<sup>N</sup>*

phases and hence *<sup>M</sup> <sup>k</sup>* MRT coefficients, the phases are *p* ¼ 0, *k*, 2*k*, …, *M* � *k*. These MRT coefficients have both positive and negative sets simultaneously.

When *k* is not a divisor of *M*, for a valid phase, only one among the positive or negative sets exists. The valid phases are an arithmetic series *p* ¼ 0, *k*, 2*k*, …, *N* � *k*. In this case, there is no integer *<sup>c</sup>* that satisfies *<sup>p</sup>* <sup>þ</sup> *ck* <sup>¼</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>M</sup>*, and hence, *<sup>Y</sup>*ð Þ *<sup>p</sup> k* ¼ �*Y*ð Þ *<sup>p</sup>*þ*ck <sup>k</sup>* cannot be true. Thus, for *p* > *M*, where *p* is a valid phase, there is no valid phase index *<sup>p</sup>* � *<sup>M</sup>*. However, the relation *<sup>Y</sup>*ð Þ *<sup>p</sup> <sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*�*<sup>M</sup> <sup>k</sup>* is still satisfied. Thus, for *p*> *M*, there is an allowable but nonvalid phase *p* � *M*. We can use the relation *Y*ð Þ *<sup>p</sup> <sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*�*<sup>M</sup> <sup>k</sup>* to express the MRT corresponding to a valid phase index *p* > *M* in terms of an allowable phase index. MRT coefficients formed by valid phases have only positive sets. Thus, the positive set of the MRT coefficient with *p*> *M* is the negative set of the MRT coefficient with allowable nonvalid phase *p* � *M*. Hence, a subset of the *M* allowable phase indices are valid phases, while the other subset is made up of allowable nonvalid phases of the form *p* � *M* that are obtained from valid phases *p* > *M*.

The gap between two successive valid phases is *k*. Let there be an allowable and numerically smallest valid phase *p*<sup>1</sup> < *M*, and let the nearest allowable nonvalid phase be *p*2. The valid phase corresponding to *p*<sup>2</sup> is *p*<sup>0</sup> <sup>2</sup> ¼ *p*<sup>2</sup> þ *M*. Let *q* be the smallest integer such that *qk*> *M*. Then,

$$p\_2' = p\_1 + qk$$

*p*<sup>2</sup> is the closest allowable nonvalid phase to *p*1. *p*<sup>1</sup> is the lowest allowable valid phase. There is no valid phase *p*<sup>0</sup> <sup>1</sup> ¼ *p*<sup>1</sup> þ *M*. The next valid phase is hence given by *p*0 <sup>2</sup> ¼ *p*<sup>1</sup> þ *qk*, since *q* is the smallest integer such that *qk*> *M*

$$\cdot \cdot p\_2 - p\_1 = qk - M$$

Since *<sup>k</sup>*∣*N*, there is an integer *<sup>t</sup>* satisfying *<sup>N</sup>* <sup>¼</sup> *tk*. Hence, *<sup>M</sup> <sup>k</sup>* <sup>¼</sup> *<sup>t</sup>* 2 . Since *k* does not divide *M*, *<sup>t</sup>* <sup>2</sup> is not an integer. For this to be true, *<sup>t</sup>* has to be odd. We know *<sup>k</sup>* <sup>¼</sup> *<sup>N</sup> t* . Since the division of any even number by an odd number produces an even number, *k* has to be even. Since *k*∣*N* and *k* is even, it follows that *<sup>k</sup>* <sup>2</sup> ∣*M*.

Hence,

$$M = \frac{dk}{2}, d \text{ being an integer}$$

$$p\_2 - p\_1 = qk - \frac{dk}{2}$$

$$= \frac{2qk}{2} - \frac{dk}{2}$$

$$= (2q - d)\frac{k}{2}$$

The value of 2ð Þ *q* � *d* cannot be larger than 1, since, in that case,

*p*<sup>2</sup> � *p*<sup>1</sup> is greater than or equal to *k:*

The distance between *p*<sup>0</sup> <sup>2</sup> and *M* has to be lesser than *k*. Thus, the distance between *p*<sup>1</sup> and *p*<sup>2</sup> has to be lesser than *k*. Hence,

$$p\_2 - p\_1 = \frac{k}{2} \tag{13}$$

0, *<sup>k</sup>*, 2*k*, ………………, *<sup>M</sup>* � *<sup>k</sup>*

and the sequence of allowable phases that correspond to MRT coefficients hav-

a. When *k*∣*N*, the index *na* of the first element in the positive data group of an

b. For *k*∣*M*, if an element with index *n* belongs in the positive data set of an MRT

*<sup>k</sup>* , when *<sup>k</sup>* does not divide *<sup>M</sup>*, is *nb* <sup>¼</sup> *<sup>p</sup>*þ*<sup>M</sup>*

ð Þ ð Þ *nk <sup>N</sup>* ¼ *p:*

Solutions exist for Eq. (4) only if *g k*ð Þ , *N* divides *p*. Since *g k*ð Þ¼ , *N k*, the former

The condition *g k*ð Þ¼ , *M k* is necessary for both positive and negative sets to be

¼ ½ � *nk <sup>N</sup>* þ

c. From (18), when *k* does not divide *M*, the sequence of allowable phase

From (19), the sequence of allowable phase indices that correspond to MRT

*M k k* 

2, ……………*:*, *<sup>M</sup>* � *<sup>k</sup>*

*k* 2, <sup>3</sup> *<sup>k</sup>* 2, <sup>5</sup> *<sup>k</sup>* 2, *:*

*<sup>k</sup>* is *na* <sup>¼</sup> *<sup>p</sup>*

*k*.

*<sup>k</sup>* , then the element with index *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

c. The first element in the positive set of an MRT coefficient *Y*ð Þ *<sup>p</sup>*

ing only negative sets is

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

**2.6 Theorem 3: index of first element**

set of the same MRT coefficient.

not divide *<sup>M</sup>*, has the index *na* <sup>¼</sup> *<sup>p</sup>*

a. The first element in MRT coefficient *Y*ð Þ *<sup>p</sup>*

Since *<sup>k</sup>*∣*p*, the smallest solution to Eq. (4) is *na* <sup>¼</sup> *<sup>p</sup>*

member in the positive data set of MRT coefficient *Y*ð Þ *<sup>p</sup>*

*n* þ *M k <sup>k</sup>* 

*k k*

coefficients having only positive groups is

*N*

*<sup>N</sup>* ¼ *p* þ *M*.

2, *<sup>k</sup>*, 3 *<sup>k</sup>*

Hence, if index *<sup>n</sup>* belongs in the positive set, index *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

0, *<sup>k</sup>*

MRT coefficient *Y*ð Þ *<sup>p</sup>*

MRT coefficient *Y*ð Þ *<sup>p</sup>*

smallest solution of (4),

condition can be written as *k*∣*p*.

present in an MRT coefficient.

b. Given ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*,

negative set since *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

indices is

**9**

**Proof.**

coefficient *Y*ð Þ *<sup>p</sup>*

<sup>2</sup> (19)

*<sup>k</sup>* occurs in the negative data

*<sup>k</sup>* when *k* does

*<sup>k</sup>* is the first

*<sup>k</sup>* will be present in the

……………, *M* � *k* (20)

*<sup>k</sup>*. The first element in the negative set of an

*<sup>k</sup>* will have an index that is the

*<sup>k</sup>*, and thus *na* <sup>¼</sup> *<sup>p</sup>*

*k* .

*N*

2

¼ *p* þ *M*

*k* .

The next valid phase after *p*<sup>1</sup> is

$$p\_3 = p\_1 + k \tag{14}$$

From Eqs. (13) and (14),

$$p\_3 - p\_2 = \frac{k}{2}$$

Let *p*<sup>0</sup> <sup>4</sup> be the next valid phase index after *p*<sup>0</sup> <sup>2</sup>, then

$$p'\_4 = p'\_2 + k \tag{15}$$

*p*0 <sup>4</sup> has a corresponding nonvalid allowable phase *p*<sup>4</sup> given by

$$p\_4 = p\_4' - M \tag{16}$$

From Eqs. (15) and (16),

$$p\_4 = p\_2 + k$$

$$\therefore p\_4 - p\_3 = \frac{k}{2}$$

Hence, there exists an allowable nonvalid phase between every consecutive pair of allowable valid phases. The MRT coefficient produced by these allowable nonvalid phases will have equal magnitude and opposite sign as the MRT coefficient produced by the corresponding non-allowable valid phases.

The sequence of allowable phase indices would thus be:

$$p\_0, p\_0 + \frac{k}{2}p\_0 + k, p\_0 + 3\frac{k}{2} \dots + \dots + \dots, p\_0 + M - \frac{k}{2} \tag{17}$$

When *k* does not divide *M*, MRT coefficients exist for these allowable phase indices and they will have either a positive group or a negative group only. There are *<sup>N</sup> <sup>k</sup>* such allowable phases, and when *k* does not divide *M*, *<sup>N</sup> <sup>k</sup>* is odd. Since *k*∣*N*, the condition for existence for solutions is *k*∣*p*. The smallest value of *p* that satisfies this condition is *p* ¼ 0. Hence, the first valid allowable phase is *p*<sup>0</sup> ¼ 0. MRT coefficients having only a positive set have allowable phases that are even multiples of *<sup>k</sup>* 2, starting from 0 and ending at *<sup>N</sup>*�*<sup>k</sup>* <sup>2</sup> . There are *<sup>N</sup>* <sup>2</sup>*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> such MRT coefficients. Similarly, the MRT coefficients having only a negative set have allowable phases that are odd multiples of *<sup>k</sup>* <sup>2</sup>, starting from *<sup>k</sup>* <sup>2</sup> and ending at *<sup>M</sup>* � *<sup>k</sup>*. There are *<sup>N</sup>* <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup> such MRT coefficients. The total number of 1-D MRT coefficients is thus *<sup>N</sup> <sup>k</sup>* , and the sequence of allowable phases is:

$$\{0, \frac{k}{2}k, \ \ 3\frac{k}{2}, \ldots, \ldots, \ldots, M-\frac{k}{2} \}\tag{18}$$

The sequence of allowable phases that correspond to MRT coefficients having only positive sets is

*A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

$$\{0, k, 2k, \ldots, \ldots, \ldots, M - \frac{k}{2}\}\tag{19}$$

and the sequence of allowable phases that correspond to MRT coefficients having only negative sets is

$$\frac{k}{2,} \ $ \frac{k}{2,} \$  \frac{k}{2,} \text{....} \frac{1}{2,} \text{....} \text{---} \text{...}, M-k \tag{20}$$

#### **2.6 Theorem 3: index of first element**


#### **Proof.**

*<sup>p</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>1</sup> <sup>¼</sup> *<sup>k</sup>*

*<sup>p</sup>*<sup>3</sup> � *<sup>p</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*

*p*0 <sup>4</sup> ¼ *p*<sup>0</sup>

*p*<sup>4</sup> ¼ *p*<sup>0</sup>

*p*<sup>4</sup> ¼ *p*<sup>2</sup> þ *k* <sup>∴</sup>*p*<sup>4</sup> � *<sup>p</sup>*<sup>3</sup> <sup>¼</sup> *<sup>k</sup>*

of allowable valid phases. The MRT coefficient produced by these allowable

produced by the corresponding non-allowable valid phases. The sequence of allowable phase indices would thus be:

2, *<sup>p</sup>*<sup>0</sup> <sup>þ</sup> *<sup>k</sup>*, *<sup>p</sup>*<sup>0</sup> <sup>þ</sup> <sup>3</sup>

*<sup>k</sup>* such allowable phases, and when *k* does not divide *M*, *<sup>N</sup>*

coefficients. The total number of 1-D MRT coefficients is thus *<sup>N</sup>*

2, *<sup>k</sup>*, 3 *<sup>k</sup>*

0, *<sup>k</sup>*

*k*

*p*0, *p*<sup>0</sup> þ

starting from 0 and ending at *<sup>N</sup>*�*<sup>k</sup>*

<sup>2</sup>, starting from *<sup>k</sup>*

Hence, there exists an allowable nonvalid phase between every consecutive pair

nonvalid phases will have equal magnitude and opposite sign as the MRT coefficient

*k* 2

When *k* does not divide *M*, MRT coefficients exist for these allowable phase indices and they will have either a positive group or a negative group only. There

condition for existence for solutions is *k*∣*p*. The smallest value of *p* that satisfies this condition is *p* ¼ 0. Hence, the first valid allowable phase is *p*<sup>0</sup> ¼ 0. MRT coefficients having only a positive set have allowable phases that are even multiples of *<sup>k</sup>*

the MRT coefficients having only a negative set have allowable phases that are odd

The sequence of allowable phases that correspond to MRT coefficients having

<sup>2</sup>*<sup>k</sup>* <sup>þ</sup> <sup>1</sup>

<sup>2</sup> and ending at *<sup>M</sup>* � *<sup>k</sup>*. There are *<sup>N</sup>*

2, ………………, *<sup>M</sup>* � *<sup>k</sup>*

<sup>2</sup> . There are *<sup>N</sup>*

2

………………, *<sup>p</sup>*<sup>0</sup> <sup>þ</sup> *<sup>M</sup>* � *<sup>k</sup>*

<sup>2</sup> (17)

*<sup>k</sup>* is odd. Since *k*∣*N*, the

<sup>2</sup> such MRT

<sup>2</sup> (18)

*<sup>k</sup>* , and the sequence

<sup>2</sup> such MRT coefficients. Similarly,

<sup>2</sup>*<sup>k</sup>* � <sup>1</sup>

2,

<sup>4</sup> has a corresponding nonvalid allowable phase *p*<sup>4</sup> given by

2

<sup>2</sup>, then

The next valid phase after *p*<sup>1</sup> is

<sup>4</sup> be the next valid phase index after *p*<sup>0</sup>

From Eqs. (13) and (14),

*Number Theory and Its Applications*

From Eqs. (15) and (16),

Let *p*<sup>0</sup>

*p*0

are *<sup>N</sup>*

multiples of *<sup>k</sup>*

of allowable phases is:

only positive sets is

**8**

<sup>2</sup> (13)

*p*<sup>3</sup> ¼ *p*<sup>1</sup> þ *k* (14)

<sup>2</sup> þ *k* (15)

<sup>4</sup> � *M* (16)

a. The first element in MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* will have an index that is the smallest solution of (4),

$$((nk))\_N = p.$$

Solutions exist for Eq. (4) only if *g k*ð Þ , *N* divides *p*. Since *g k*ð Þ¼ , *N k*, the former condition can be written as *k*∣*p*.

Since *<sup>k</sup>*∣*p*, the smallest solution to Eq. (4) is *na* <sup>¼</sup> *<sup>p</sup> <sup>k</sup>*, and thus *na* <sup>¼</sup> *<sup>p</sup> <sup>k</sup>* is the first member in the positive data set of MRT coefficient *Y*ð Þ *<sup>p</sup> k* .

The condition *g k*ð Þ¼ , *M k* is necessary for both positive and negative sets to be present in an MRT coefficient.

b. Given ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*,

$$\left\| \left[ \left( n + \frac{M}{k} \right) k \right] \right\|\_{N} = [nk]\_{N} + \left\| \frac{M}{k} k \right\|\_{N} = p + M$$

Hence, if index *<sup>n</sup>* belongs in the positive set, index *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup> <sup>k</sup>* will be present in the negative set since *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup> k k <sup>N</sup>* ¼ *p* þ *M*.

c. From (18), when *k* does not divide *M*, the sequence of allowable phase indices is

$$0, \frac{k}{2}k, \ 3\frac{k}{2}, \ldots, \ldots, \ldots, M - \frac{k}{2}$$

From (19), the sequence of allowable phase indices that correspond to MRT coefficients having only positive groups is

$$\{0, k, 2k, \ldots, \ldots, \ldots, \ldots, M - \frac{k}{2}\}$$

*p* ¼ 0, *k*, 2*k*, …, *M* � *k* (23)

þ ……*::* þ *xp*

2

þ ……*::* þ *xp*

*k* þð Þ *<sup>k</sup>*�<sup>1</sup> *<sup>N</sup> k*

*k*þð Þ *<sup>k</sup>*�<sup>1</sup> *<sup>N</sup> k*

*<sup>k</sup>* . An MRT coefficient with

(24)

(25)

(26)

When *k* does not divide *M*, it has been seen that positive and negative groups cannot exist together for the same MRT coefficient. For certain values of *p*, only positive groups exist. For other values of *p*, only negative groups exist. As seen, for

h i

*xjN*þ*<sup>p</sup> k* h i

Similarly, an MRT coefficient with only a negative group has the following form:

� �

*xjN*þ*p*þ*<sup>M</sup> k* h i

*<sup>p</sup>* <sup>¼</sup> 0, *<sup>k</sup>*, 2*k*, …, *<sup>M</sup>* � *<sup>k</sup>*

*<sup>k</sup>*, and for negative groups, *nb* <sup>¼</sup> *<sup>p</sup>*þ*<sup>M</sup>*

*j*¼0

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* <sup>¼</sup> <sup>X</sup> *k*�1

positive groups, *na* <sup>¼</sup> *<sup>p</sup>*

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

only a positive group has the following form:

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* ¼ *xp k* þ *xp k*þ*<sup>N</sup> k* þ *xp k*þ2*<sup>N</sup> k* þ *xp k*þ3*<sup>N</sup> k*

*Y*ð Þ *<sup>p</sup>*

**2.8 Physical significance**

**11**

which, when simplified, becomes

*<sup>k</sup>* ¼ � *xp*þ*<sup>M</sup>*

*k*

method. The MRT coefficient has the following form

þ *xp*þ*<sup>M</sup> <sup>k</sup>* <sup>þ</sup>*<sup>N</sup> k* þ *xp*þ*<sup>M</sup> <sup>k</sup>* <sup>þ</sup>2*<sup>N</sup> k*

*Y*ð Þ *<sup>p</sup>*

*<sup>p</sup>* <sup>¼</sup> *<sup>k</sup>* 2 , 3*k*

*Y*ð Þ *<sup>p</sup>*

coefficient *Yk* can be expressed in terms of associated MRT as

*<sup>k</sup> <sup>W</sup>*<sup>0</sup>

<sup>8</sup> <sup>þ</sup> *<sup>Y</sup>*ð Þ<sup>1</sup> *<sup>k</sup> <sup>W</sup>*<sup>1</sup>

*Yk* <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup>

*<sup>k</sup>* ¼ �<sup>X</sup> *k*�1

*j*¼0

When *k* and *N* are co-prime, *g k*ð Þ¼ , *N* 1. Hence, the positive set has only one element and similarly the negative set too has only one element. The values of *na* and *nb* have to be computed using the Euclidean algorithm or by the trial-and-error

> *<sup>k</sup>* ¼ *xna* � *xnb p* ¼ 0, 1, 2, 3, … *M* � 1

An MRT coefficient possesses both frequency and phase. A DFT coefficient has only the frequency index. Hence, the presence of an extra index distinguishes the MRT coefficient. The physical significance of an MRT coefficient is that the phase specifies the beginning of the frequency cycle. The MRT can thus be thought of as a time-frequency representation of a 1-D signal. In contrast to the DFT, the MRT while related to the DFT, has localization in both time and frequency. Also, MRT coefficients can be considered to be constituent parts of the DFT; these parts, if weighted by the exponential kernel, would produce the DFT. For *N* ¼ 8, the DFT

> <sup>8</sup> <sup>þ</sup> *<sup>Y</sup>*ð Þ<sup>2</sup> *<sup>k</sup> <sup>W</sup>*<sup>2</sup>

<sup>8</sup> <sup>þ</sup> *<sup>Y</sup>*ð Þ<sup>2</sup> *<sup>k</sup> <sup>W</sup>*<sup>3</sup> 8

<sup>2</sup> , …, *<sup>M</sup>* � *<sup>k</sup>*

which, when simplified, becomes

From (20), the sequence of allowable phase indices that correspond to MRT coefficients having only negative groups is

$$\frac{k}{2,} 3 \frac{k}{2,} 5 \frac{k}{2,} \dots = \dots = \dots, M - k$$

The first data element that satisfies ð Þ *nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* is *<sup>p</sup> <sup>k</sup>* since *k*∣*p* is satisfied, *p* being a valid phase index. Hence, the first element in the positive set has index *na* <sup>¼</sup> *<sup>p</sup> k*.

Also, the allowable phase indices that produce MRT coefficients having only negative sets are actually nonvalid allowable phase indices. If *pb* is a nonvalid phase index, there is a valid phase index *pa* given by *pa* ¼ *pb* þ *M*. Hence, the first data element in the negative set has the index *nb* <sup>¼</sup> *<sup>p</sup>*þ*<sup>M</sup> k* .

If *<sup>k</sup>* does not divide *<sup>N</sup>*, then *na* <sup>¼</sup> *<sup>p</sup> g k*ð Þ , *<sup>N</sup>* cannot be a solution since *g k*ð Þ , *<sup>N</sup>* <sup>∣</sup>*k*. The extended Euclidean algorithm can be used to find a particular solution for this case.

#### **2.7 Closed-form expression for 1-D MRT**

By the MRT definition, and looking at element indices of both positive and negative data sets in (12), and, assuming *na* belongs in the positive set, and *nb* in the negative set, a 1-D MRT coefficient can be expressed in the following manner:

$$Y\_k^{(p)} = \left[\mathbf{x}\_{n\_a} + \mathbf{x}\_{n\_d + \mathfrak{g}\_k} + \mathbf{x}\_{n\_a + 2\mathfrak{g}\_k} + \mathbf{x}\_{n\_a + 3\mathfrak{g}\_k} + \dots + \mathbf{x}\_{n\_d + \mathfrak{g}(k, \mathcal{N}) - \mathfrak{g}\_k}\right]$$

$$-\left[\mathbf{x}\_{n\_b} + \mathbf{x}\_{n\_b + \mathfrak{g}\_k} + \mathbf{x}\_{n\_b + 2\mathfrak{g}\_k} + \mathbf{x}\_{n\_b + 3\mathfrak{g}\_k} + \dots + \dots + \mathbf{x}\_{n\_b + \mathfrak{g}(k, \mathcal{N}) - \mathfrak{g}\_k}\right]$$

This can be written as

$$Y\_{k\_{(p)}} = \sum\_{j=0}^{g(k,N)-1} \varkappa\_{n\_a + j\mathfrak{g}\_k} - \varkappa\_{n\_b + j\mathfrak{g}\_k} \tag{21}$$

Here, Eq. (21) is a closed-form formula for MRT coefficients.

When *k* divides *N*, using Theorem 3(a), we observe that the first element in the positive set is *na* <sup>¼</sup> *<sup>p</sup> k* . Also, using Theorem 3(b), any index in the positive set is related to any index in the negative set, given *<sup>k</sup>*∣*M*. This relation is *nb* <sup>¼</sup> *na* <sup>þ</sup> *<sup>M</sup> <sup>k</sup>* When *k* does not divide *M*, from Theorem 3(c), we obtain the phases that correspond to MRT coefficients with only positive sets, and MRT coefficients with only negative sets.

When *<sup>k</sup>*∣*M*, *na* <sup>¼</sup> *<sup>p</sup> <sup>k</sup>*, and *nb* <sup>¼</sup> *na* <sup>þ</sup> *<sup>M</sup> <sup>k</sup>* . Also, *k* does not divide *M*. Using these, we can rewrite Eq. (21) as,

$$\begin{split} Y\_k^{(p)} &= \left[ \mathbf{x}\_{\frac{p}{k}} + \mathbf{x}\_{\frac{p}{k} + \frac{N}{k}} + \mathbf{x}\_{\frac{p}{k} + \frac{2N}{k}} + \mathbf{x}\_{\frac{p}{k} + \frac{3N}{k}} + \dots \dots + \mathbf{x}\_{\frac{p}{k} + \frac{(k-1)N}{k}} \right] \\ &- \left[ \mathbf{x}\_{\frac{p}{k} + \frac{N}{2k}} + \mathbf{x}\_{\frac{p}{k} + \frac{3N}{2k}} + \mathbf{x}\_{\frac{p}{k} + \frac{5N}{2k}} + \mathbf{x}\_{\frac{p}{k} + \frac{7N}{2k}} + \dots \dots + \mathbf{x}\_{\frac{p}{k} + \frac{(2k-1)N}{2k}} \right] \end{split} \tag{22}$$

On further simplification,

$$Y\_k^{(p)} = \sum\_{j=0}^{k-1} \left[ \mathfrak{x}\_{\frac{jN+p}{k}} - \mathfrak{x}\_{\frac{(2j+1)N+2p}{2k}} \right]$$

*A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

0, *<sup>k</sup>*, 2*k*, …………………, *<sup>M</sup>* � *<sup>k</sup>*

From (20), the sequence of allowable phase indices that correspond to MRT

valid phase index. Hence, the first element in the positive set has index *na* <sup>¼</sup> *<sup>p</sup>*

Also, the allowable phase indices that produce MRT coefficients having only negative sets are actually nonvalid allowable phase indices. If *pb* is a nonvalid phase index, there is a valid phase index *pa* given by *pa* ¼ *pb* þ *M*. Hence, the first data

extended Euclidean algorithm can be used to find a particular solution for this case.

By the MRT definition, and looking at element indices of both positive and negative data sets in (12), and, assuming *na* belongs in the positive set, and *nb* in the negative set, a 1-D MRT coefficient can be expressed in the following manner:

> *<sup>k</sup>* ¼ *xna* þ *xna*þ*gk* þ *xna*þ2*gk* þ *xna*þ3*gk* þ *::*…… þ *xna*þ*g k*ð Þ� ,*<sup>N</sup>* <sup>1</sup>*gk* h i

> > *g k*ð Þ� X, *<sup>N</sup>* <sup>1</sup>

> > > *j*¼0

When *k* divides *N*, using Theorem 3(a), we observe that the first element in the

h i

*xjN*þ*<sup>p</sup> k*

not divide *M*, from Theorem 3(c), we obtain the phases that correspond to MRT coefficients with only positive sets, and MRT coefficients with only negative sets.

. Also, using Theorem 3(b), any index in the positive set is related

þ ……*::* þ *xp*

h i (22)

� *x*ð Þ <sup>2</sup>*j*þ<sup>1</sup> *<sup>N</sup>*þ2*<sup>p</sup>* 2*k*

h i

Here, Eq. (21) is a closed-form formula for MRT coefficients.

to any index in the negative set, given *<sup>k</sup>*∣*M*. This relation is *nb* <sup>¼</sup> *na* <sup>þ</sup> *<sup>M</sup>*

*Yk*ð Þ *<sup>p</sup>* <sup>¼</sup>

*<sup>k</sup>*, and *nb* <sup>¼</sup> *na* <sup>þ</sup> *<sup>M</sup>*

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* <sup>¼</sup> <sup>X</sup> *k*�1

*j*¼0

� *xnb* þ *xnb*þ*gk* þ *xnb*þ2*gk* þ *xnb*þ3*gk* þ ……… þ *xnb*þ*g k*ð Þ� ,*<sup>N</sup>* <sup>1</sup>*gk* h i

……………, *M* � *k*

*k* .

*g k*ð Þ , *<sup>N</sup>* cannot be a solution since *g k*ð Þ , *<sup>N</sup>* <sup>∣</sup>*k*. The

*xna*þ*jgk* � *xnb*þ*jgk* (21)

*<sup>k</sup>* . Also, *k* does not divide *M*. Using these, we

*k*þð Þ *<sup>k</sup>*�<sup>1</sup> *<sup>N</sup> k*

> *k*þð Þ <sup>2</sup>*k*�<sup>1</sup> *<sup>N</sup>* 2*k*

þ ……*::* þ *xp*

coefficients having only negative groups is

*Number Theory and Its Applications*

*k* 2, <sup>3</sup> *<sup>k</sup>* 2, <sup>5</sup> *<sup>k</sup>* 2, *:*

The first data element that satisfies ð Þ *nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* is *<sup>p</sup>*

element in the negative set has the index *nb* <sup>¼</sup> *<sup>p</sup>*þ*<sup>M</sup>*

If *<sup>k</sup>* does not divide *<sup>N</sup>*, then *na* <sup>¼</sup> *<sup>p</sup>*

**2.7 Closed-form expression for 1-D MRT**

*Y*ð Þ *<sup>p</sup>*

positive set is *na* <sup>¼</sup> *<sup>p</sup>*

When *<sup>k</sup>*∣*M*, *na* <sup>¼</sup> *<sup>p</sup>*

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* ¼ *xp k* þ *xp k*þ*<sup>N</sup> k* þ *xp k*þ2*<sup>N</sup> k* þ *xp k*þ3*<sup>N</sup> k*

On further simplification,

� *xp k*þ*<sup>N</sup>* 2*k* þ *xp k*þ3*<sup>N</sup>* 2*k* þ *xp k*þ5*<sup>N</sup>* 2*k* þ *xp k*þ7*<sup>N</sup>* 2*k*

can rewrite Eq. (21) as,

**10**

This can be written as

*k*

2

*<sup>k</sup>* since *k*∣*p* is satisfied, *p* being a

*k*.

*<sup>k</sup>* When *k* does

$$p = 0, k, 2k, \ldots, M - k \tag{23}$$

When *k* does not divide *M*, it has been seen that positive and negative groups cannot exist together for the same MRT coefficient. For certain values of *p*, only positive groups exist. For other values of *p*, only negative groups exist. As seen, for positive groups, *na* <sup>¼</sup> *<sup>p</sup> <sup>k</sup>*, and for negative groups, *nb* <sup>¼</sup> *<sup>p</sup>*þ*<sup>M</sup> <sup>k</sup>* . An MRT coefficient with only a positive group has the following form:

$$Y\_k^{(p)} = \left[ \mathfrak{x}\_{\frac{\mathfrak{p}}{k}} + \mathfrak{x}\_{\frac{\mathfrak{p}}{k} + \frac{\mathfrak{q}}{k}} + \mathfrak{x}\_{\frac{\mathfrak{p}}{k} + \frac{\mathfrak{q}\mathfrak{q}}{k}} + \mathfrak{x}\_{\frac{\mathfrak{p}}{k} + \frac{\mathfrak{q}\mathfrak{q}}{k}} + \dots \dots + \mathfrak{x}\_{\frac{\mathfrak{p}}{k} + \frac{(k-1)\mathfrak{q}}{k}} \right]$$

which, when simplified, becomes

$$\begin{aligned} Y\_k^{(p)} &= \sum\_{j=0}^{k-1} \left[ \varkappa\_{\frac{N+p}{k}} \right] \\ p &= 0, \ k, \ 2k, \ldots, M - \frac{k}{2} \end{aligned} \tag{24}$$

Similarly, an MRT coefficient with only a negative group has the following form:

$$Y\_k^{(p)} = -\left[\mathfrak{x}\_{\frac{p+M}{k}} + \mathfrak{x}\_{\frac{p+M}{k} + \frac{N}{k}} + \mathfrak{x}\_{\frac{p+M}{k} + \frac{2N}{k}} + \dots + \mathfrak{x}\_{\frac{p+(k-1)N}{k}}\right]^T$$

which, when simplified, becomes

$$\begin{aligned} Y\_k^{(p)} &= -\sum\_{j=0}^{k-1} \left[ \mathfrak{X}\_{\frac{jN+p+M}{k}} \right] \\ p &= \frac{k}{2}, \frac{3k}{2}, ..., M-k \end{aligned} \tag{25}$$

When *k* and *N* are co-prime, *g k*ð Þ¼ , *N* 1. Hence, the positive set has only one element and similarly the negative set too has only one element. The values of *na* and *nb* have to be computed using the Euclidean algorithm or by the trial-and-error method. The MRT coefficient has the following form

$$\begin{aligned} Y\_k^{(p)} &= \boldsymbol{\varkappa}\_{n\_a} - \boldsymbol{\varkappa}\_{n\_b} \\ p &= 0, 1, 2, 3, \dots \ M - 1 \end{aligned} \tag{26}$$

#### **2.8 Physical significance**

An MRT coefficient possesses both frequency and phase. A DFT coefficient has only the frequency index. Hence, the presence of an extra index distinguishes the MRT coefficient. The physical significance of an MRT coefficient is that the phase specifies the beginning of the frequency cycle. The MRT can thus be thought of as a time-frequency representation of a 1-D signal. In contrast to the DFT, the MRT while related to the DFT, has localization in both time and frequency. Also, MRT coefficients can be considered to be constituent parts of the DFT; these parts, if weighted by the exponential kernel, would produce the DFT. For *N* ¼ 8, the DFT coefficient *Yk* can be expressed in terms of associated MRT as

$$Y\_k = Y\_k^{(0)} W\_8^0 + Y\_k^{(1)} W\_8^1 + Y\_k^{(2)} W\_8^2 + Y\_k^{(2)} W\_8^3$$

#### **3. Redundancy in MRT**

For some values of *N*, a set of MRT coefficients having different values for frequency and phase have the same magnitude. The polarity of the coefficients may be different. This phenomenon is an indication of redundancy in MRT. The MRT can be relieved of the redundancy to arrive at a simpler transform that has no redundancy.

#### **3.1 Theorem 4: complete redundancy**

Given *Y*ð Þ *<sup>p</sup> <sup>k</sup>* , for all *h* such that *g h*ð Þ¼ , *N* 1

$$\begin{aligned} Y\_{(hk)\_N}^{\left( (hp)\_N \right)} &= Y\_k^{(p)} \text{ for } [hp]\_N < M \text{, and},\\ Y\_{(hk)\_N}^{\left( (hp)\_N - M \right)} &= -Y\_k^{(p)} \text{ for } [hp]\_N \ge M \end{aligned}$$

**Proof.** From a basic theorem [16] in number theory, if

$$[q]\_N = d \tag{27}$$

From the definition of MRT, and Eqs. (32) and (36), it is seen that the set of

*<sup>Y</sup>* ð Þ *hp* ð Þ*<sup>N</sup>* ð Þ *hk <sup>N</sup>*

*<sup>Y</sup>* ð Þ *hp* ð Þ *<sup>N</sup>*�*<sup>M</sup>* ð Þ *hk <sup>N</sup>*

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*�*<sup>M</sup>*

*k* ¼ 5 are completely redundant MRT coefficients of *k* ¼ 1, since 1 and 5 are

obtain *k*<sup>0</sup> by multiplying an integer *k* with an integer *h*, *k*<sup>0</sup> ¼ *hk*, satisfying *g h*ð Þ¼ , *N* 1 and *k* 6¼ *k*<sup>0</sup> Thus, MRT coefficients with frequency indices that are

nontrivial common divisor with *N*. So, even though it is not a divisor of *N*, it will

MRT coefficients with frequency indices that have common gcd w.r.t. *N* are all

**Proof.** Assume *k*<sup>1</sup> and *k*<sup>2</sup> are two frequency indices that have common gcd

From Theorem 4, we see that prediction of redundancy is possible from the knowledge of *N*, frequency and phase. Given ð Þ *k*, *p* , frequency and phase pairs of redundant MRT coefficients are given by Theorem 4. The redundancy condition is that the factor of multiplication that connects the frequencies of redundant MRT coefficients should be co-prime to *N*. As an example, for *N* ¼ 6, MRT coefficients of

If *k*<sup>0</sup> ¼ *hk* and *g h*ð Þ¼ , *N* 1, then MRT coefficients of the two frequency indices *k*

When *k* ¼ 1, MRT coefficients having frequency *k*<sup>0</sup> that are co-prime to *N* can be

Hence, the theorem on complete redundancy has been proved.

*Y*ð Þ *<sup>p</sup>*

obtained from MRT coefficients with a frequency of *k* ¼ 1. If *k*<sup>0</sup>

Let k<sup>0</sup> satisfy *k*<sup>0</sup> ¼ *hk* such that *g h*ð Þ¼ , *N* 1 and *k* 6¼ *k*<sup>0</sup>

have complete redundancy with the common divisor.

divisors of *N* cannot be redundant to each other.

**3.2 Theorem 5: redundancy frequency groups**

From Eq. (41), *h* and *N* are co-prime. Hence,

completely redundant to each other.

Assume *h* exists such that

ð Þ *hk <sup>N</sup>*

<sup>¼</sup> *<sup>Y</sup>*ð Þ *<sup>p</sup>*

¼ �*Y*ð Þ *<sup>p</sup>*

*<sup>k</sup>* ¼ �*Y*ð Þ *<sup>p</sup>*þ*<sup>M</sup> k*

. Using Eqs. (30)–(32) and (36),

*<sup>k</sup>* (37)

*<sup>k</sup>* (38)

∣*N*, then we cannot

. If *k*<sup>0</sup> is nonprime, there is a

*g k*ð Þ¼ 1, *N k* (39) *g k*ð Þ¼ 2, *N k* (40)

*g h*ð Þ¼ , *N* 1 (41)

*g hk* ð Þ¼ 1, *N k* (42)

½ �� ½*hk*<sup>1</sup> *<sup>N</sup>* ¼ *hk*<sup>1</sup> � *Nq* if 0≤ *hk*<sup>1</sup> � *Nq* < *N*, *q*∈ (43)

indices *n* and *n*<sup>0</sup> form the MRT coefficient *Y*ð Þ *hp <sup>N</sup>*

If ð Þ *hp <sup>N</sup>* ≥ *M*, then

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

since

co-prime to 6.

w.r.t. *N*.

**13**

and *k*<sup>0</sup> are redundant.

and *h* is a multiplication factor, then

$$\left[\left[hq\right]\_{\frac{N}{k(h,N)}}=hd\right] \tag{28}$$

If *g h*ð Þ¼ , *N* 1, Eq. (28) becomes

$$\|\|hq\|\|\_{N} = hd \tag{29}$$

Given *Y*ð Þ *<sup>p</sup> <sup>k</sup>* , and a set of indices *n* that satisfies

$$\|\boldsymbol{n}k\|\_{N} = p \tag{30}$$

and a set of indices *n*<sup>0</sup> that satisfies

$$\|\boldsymbol{n}'\boldsymbol{k}\|\_{N} = \boldsymbol{p} + \boldsymbol{M} \tag{31}$$

If there is *h* such that *g h*ð Þ¼ , *N* 1, using Eqs. (27)–(29),

$$\|\|n(hk)\|\|\_{N} = hp \tag{32}$$

and

$$\|\|n'(hk)\|\|\_{N} = h(p+M) \tag{33}$$

$$\|n'(hk)\|\_{N} = hp + hM \tag{34}$$

Since *g h*ð Þ¼ , *N* 1, *h* is odd, and hence

$$hp + hM \equiv hp + M\tag{35}$$

Using Eq. (35), Eq. (34) may be written as

$$\|n'(hk)\|\_{N} = hp + M \tag{36}$$

From the definition of MRT, and Eqs. (32) and (36), it is seen that the set of indices *n* and *n*<sup>0</sup> form the MRT coefficient *Y*ð Þ *hp <sup>N</sup>* ð Þ *hk <sup>N</sup>* . Using Eqs. (30)–(32) and (36),

$$\left(Y\_{(hk)\_N}^{((hp)\_N)}\right) = Y\_k^{(p)}\tag{37}$$

If ð Þ *hp <sup>N</sup>* ≥ *M*, then

$$Y\_{(hk)\_N}^{\left( (hp)\_N - M \right)} = -Y\_k^{(p)} \tag{38}$$

since

**3. Redundancy in MRT**

*Number Theory and Its Applications*

**3.1 Theorem 4: complete redundancy**

and *h* is a multiplication factor, then

If *g h*ð Þ¼ , *N* 1, Eq. (28) becomes

and a set of indices *n*<sup>0</sup> that satisfies

Since *g h*ð Þ¼ , *N* 1, *h* is odd, and hence

Using Eq. (35), Eq. (34) may be written as

*<sup>k</sup>* , for all *h* such that *g h*ð Þ¼ , *N* 1

*<sup>Y</sup>* ð Þ *hp* ð Þ*<sup>N</sup>* ð Þ *hk <sup>N</sup>*

*<sup>Y</sup>* ð Þ *hp* ð Þ *<sup>N</sup>*�*<sup>M</sup>* ð Þ *hk <sup>N</sup>*

**Proof.** From a basic theorem [16] in number theory, if

*<sup>k</sup>* , and a set of indices *n* that satisfies

If there is *h* such that *g h*ð Þ¼ , *N* 1, using Eqs. (27)–(29),

<sup>¼</sup> *<sup>Y</sup>*ð Þ *<sup>p</sup>*

¼ �*Y*ð Þ *<sup>p</sup>*

½ �� ½*hq <sup>N</sup> g h*ð Þ ,*N*

*<sup>k</sup>* for½ � *hp <sup>N</sup>* < *M*, and,

*<sup>k</sup>* for½ � *hp <sup>N</sup>* ≥ *M*

½ �� ½*q <sup>N</sup>* ¼ *d* (27)

½ �� ½*hq <sup>N</sup>* ¼ *hd* (29)

½ �� ½*nk <sup>N</sup>* ¼ *p* (30)

½*n*<sup>0</sup> ½ �� *k <sup>N</sup>* ¼ *p* þ *M* (31)

½ �� ½*n hk* ð Þ *<sup>N</sup>* ¼ *hp* (32)

½*n*<sup>0</sup> ½ �� ð Þ *hk <sup>N</sup>* ¼ *h p*ð Þ þ *M* (33) ½*n*<sup>0</sup> ½ �� ð Þ *hk <sup>N</sup>* ¼ *hp* þ *hM* (34)

*hp* þ *hM* � *hp* þ *M* (35)

½*n*<sup>0</sup> ½ �� ð Þ *hk <sup>N</sup>* ¼ *hp* þ *M* (36)

¼ *hd* (28)

redundancy.

Given *Y*ð Þ *<sup>p</sup>*

Given *Y*ð Þ *<sup>p</sup>*

and

**12**

For some values of *N*, a set of MRT coefficients having different values for frequency and phase have the same magnitude. The polarity of the coefficients may be different. This phenomenon is an indication of redundancy in MRT. The MRT can be relieved of the redundancy to arrive at a simpler transform that has no

$$Y\_k^{(p)} = -Y\_k^{(p-\mathcal{M})} = -Y\_k^{(p+\mathcal{M})}$$

Hence, the theorem on complete redundancy has been proved.

From Theorem 4, we see that prediction of redundancy is possible from the knowledge of *N*, frequency and phase. Given ð Þ *k*, *p* , frequency and phase pairs of redundant MRT coefficients are given by Theorem 4. The redundancy condition is that the factor of multiplication that connects the frequencies of redundant MRT coefficients should be co-prime to *N*. As an example, for *N* ¼ 6, MRT coefficients of *k* ¼ 5 are completely redundant MRT coefficients of *k* ¼ 1, since 1 and 5 are co-prime to 6.

If *k*<sup>0</sup> ¼ *hk* and *g h*ð Þ¼ , *N* 1, then MRT coefficients of the two frequency indices *k* and *k*<sup>0</sup> are redundant.

When *k* ¼ 1, MRT coefficients having frequency *k*<sup>0</sup> that are co-prime to *N* can be obtained from MRT coefficients with a frequency of *k* ¼ 1. If *k*<sup>0</sup> ∣*N*, then we cannot obtain *k*<sup>0</sup> by multiplying an integer *k* with an integer *h*, *k*<sup>0</sup> ¼ *hk*, satisfying *g h*ð Þ¼ , *N* 1 and *k* 6¼ *k*<sup>0</sup> Thus, MRT coefficients with frequency indices that are divisors of *N* cannot be redundant to each other.

Let k<sup>0</sup> satisfy *k*<sup>0</sup> ¼ *hk* such that *g h*ð Þ¼ , *N* 1 and *k* 6¼ *k*<sup>0</sup> . If *k*<sup>0</sup> is nonprime, there is a nontrivial common divisor with *N*. So, even though it is not a divisor of *N*, it will have complete redundancy with the common divisor.

#### **3.2 Theorem 5: redundancy frequency groups**

MRT coefficients with frequency indices that have common gcd w.r.t. *N* are all completely redundant to each other.

**Proof.** Assume *k*<sup>1</sup> and *k*<sup>2</sup> are two frequency indices that have common gcd w.r.t. *N*.

$$\lg(k\_1, N) = k \tag{39}$$

$$\mathbf{g}(k\_2, N) = k \tag{40}$$

Assume *h* exists such that

$$\mathbf{g}(h, N) = \mathbf{1} \tag{41}$$

From Eq. (41), *h* and *N* are co-prime. Hence,

$$\mathbf{g}(hk\_1, N) = k \tag{42}$$

$$\|hk\_1\|\_N = hk\_1 - Nq \text{ if } 0 \le hk\_1 - Nq < N, q \in \mathbb{Z} \tag{43}$$

Using Eq. (43) and gcd property,

$$\mathbf{g}\left((hk\_1)\_N, N\right) = \mathbf{g}(hk\_1 - Nq, N) = \mathbf{g}(hk\_1, N) = k \tag{44}$$

From Eqs. (40) and (44),

$$k\_2 = [hk\_1]\_N \tag{45}$$

From Eq. (47), the sum of the terms *Φ <sup>N</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

*<sup>d</sup>*∣*NΦ <sup>N</sup> d*

**3.3 Theorem 6: Mapping between phase indices**

frequencies *k* and *k*<sup>0</sup> connected through complete redundancy.

*<sup>d</sup>*∣*NΦ*ð Þ¼ *<sup>d</sup>* <sup>P</sup>

*A New Integer-to-Integer Transform*

equal since *g k*ð Þ¼ , *N g k*<sup>0</sup>

through complete redundancy.

*p*<sup>0</sup> ¼ ½ � 0, 3, 2, 1 .

For *N* ¼ 6,

Let *Y*ð Þ *<sup>p</sup>*

*p* and *nk*<sup>0</sup> � � � �

**15**

*<sup>k</sup>*<sup>0</sup> and *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup>*

**3.4 Derived redundancy**

To look at an example, when *N* ¼ 8,

have been mapped.

since P

gcd w.r.t. *N*.

*k*

Thus, all the frequency indices of 1-D MRT can be classified on the basis of their

One-to-one mapping exists between phases of 1-D MRT coefficients having

**Proof.** For two MRT coefficients having frequencies *k* and *k*<sup>0</sup> connected by complete redundancy, the number of phases corresponding to each frequency is

The phases lie in the range 0, ½ � *M* � 1 . From Theorem 4 on complete redundancy, the relation between phases is given by *p*<sup>0</sup> ¼ ½½ �� *hp <sup>N</sup>*, *g h*ð Þ¼ , *N* 1. Using a theorem on the reduced residue systems, on multiplication with *h*, the resulting group of phase indices *p*<sup>0</sup> too have the same elements as the original group. Multiplication of phases in 0, ½ � *M* � 1 by *h* and then computing modulo w.r.t. *M* produces the same group, but with the order possibly altered. Thus, one-to-one mapping exists

between the phases of 1-D MRT coefficients having frequencies *k* and *k*<sup>0</sup> connected

*Y*ð Þ <sup>0</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>3</sup> ,

*Y*ð Þ<sup>1</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ<sup>3</sup> 3

*Y*ð Þ<sup>2</sup>

*Y*ð Þ<sup>3</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ<sup>1</sup> 3 *:*

*Y*ð Þ <sup>0</sup> <sup>3</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup>

we can be obtain it by combining other MRT coefficients.

are ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa* and ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa* <sup>þ</sup> *<sup>M</sup>*. The congruence relations for *<sup>Y</sup>*ð Þ *<sup>p</sup>*

<sup>1</sup> ¼ �*Y*ð Þ<sup>2</sup>

result in*p*<sup>0</sup> ¼ ½ � 0, 3, 6, 1 , which reduces to *p*<sup>0</sup> ¼ ½ � 0, 3, 2, 1 after the condition ½ �� ½*hp <sup>N</sup>* < *M* is checked and relevant sign change. Hence, *p* ¼ ½ � 0, 1, 2, 3 maps to

<sup>3</sup> , and,

Since *k* ¼ 1 and *k* ¼ 3 are redundant through the co-prime *h* ¼ 3, the set of phase indices of *k* ¼ 1, *p* ¼ ½ � 0, 1, 2, 3 , when subjected to the operation *p*<sup>0</sup> ¼ ½ � *hp <sup>N</sup>* would

<sup>1</sup> � *<sup>Y</sup>*ð Þ<sup>1</sup>

Certain MRT coefficients can be obtained using the summation of certain other unique MRT coefficients. This phenomenon is also a kind of redundancy, but it cannot be considered as complete redundancy. This phenomenon is referred to as derived redundancy. An MRT coefficient is considered a derived MRT coefficient if

*<sup>N</sup>* ¼ *p* þ *M*. Assume that a relation *k*<sup>0</sup> ¼ *dk* exists between *k* and *k*<sup>0</sup>

<sup>1</sup> <sup>þ</sup> *<sup>Y</sup>*ð Þ<sup>2</sup> 1

*<sup>k</sup>* be the two MRT coefficients. The congruence relations for *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup>*

� �. Hence, all the *<sup>N</sup>* frequency indices *<sup>k</sup>* <sup>¼</sup> ½ � 0, *<sup>N</sup>* � <sup>1</sup>

, *N* � �, and the number of phase indices is given by *<sup>N</sup>*

� � over all divisors of *N* is given by *N*,

*g k*ð Þ , *<sup>N</sup>* .

*k*

*<sup>N</sup>* ¼

.

*<sup>k</sup>*<sup>0</sup> are *nk*<sup>0</sup> � � �

From Eqs. (41) and (45), and using Theorem 4, it can be concluded that frequency indices *k*<sup>1</sup> and *k*<sup>2</sup> are completely redundant. Hence, the theorem is proved.

For example, if *N* ¼ 8, the following relations exist:

$$\begin{aligned} Y\_1^{(0)} &= Y\_3^{(0)} = Y\_5^{(0)} = Y\_7^{(0)} \\ Y\_1^{(1)} &= Y\_3^{(3)} = -Y\_5^{(1)} = -Y\_7^{(3)}, \\ Y\_1^{(2)} &= -Y\_3^{(2)} = Y\_5^{(2)} = -Y\_7^{(2)} \text{ and}, \\ Y\_1^{(3)} &= Y\_3^{(1)} = -Y\_5^{(3)} = -Y\_7^{(1)}. \end{aligned}$$

Since *k* ¼ 1, 3, 5 and 7 share the same gcd of 1 w.r.t. *N*, there is redundancy among the MRT coefficients of these frequencies. Similarly, since *g*ð Þ¼ 2, 8 *<sup>g</sup>*ð Þ¼ 6, 8 2, *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>2</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>6</sup> , and *<sup>Y</sup>*ð Þ<sup>2</sup> <sup>2</sup> ¼ �*Y*ð Þ<sup>2</sup> 6 .

Theorem 5 shows that we can group frequencies based on their gcd w.r.t. *N*. AII nondivisor frequency indices are related to divisor frequency indices through multiplication factors *h* such that *g h*ð Þ¼ , *N* 1. The count of the possible factors of multiplication involved in complete redundancy is Euler's totient function *Φ*ð Þ *N* , which specifies the count of positive integers smaller than or equal to *N* that are coprime to *N*, 1 being considered co-prime to every other integer. Given 1-D MRT coefficients having frequency *k* ¼ 1, there would be *Φ*ð Þ� *N* 1 other frequencies whose MRT coefficients can be obtained from this MRT coefficient. Combined with *k* ¼ 1, these *Φ*ð Þ *N* frequencies thus comprise the group of frequencies that satisfy *g h*ð Þ¼ , *N* 1. There are other sets of frequency indices that have a common gcds w.r.t. *N*.; and each group corresponds to a some divisor of *N*. There are *Φ*ð Þ *N* possible factors of multiplication that produce other members of the group of frequency indices associated with *k*. From Theorem 4, the equation for complete redundancy is *k*<sup>0</sup> ¼ ð Þ *hk <sup>N</sup>*, where *g h*ð Þ¼ , *N* 1. Also,

$$[hk]\_N = \left[ \left( h + \frac{N}{k} \right) k \right]\_N \tag{46}$$

Eq. (46) implies that the set of *k*<sup>0</sup> that is generated from divisor *k* is unique only for multiplicative factors in the set 0, *<sup>N</sup> <sup>k</sup>* � <sup>1</sup> � �, and repeats thereafter for the remaining sets of the same length. Hence, the problem now gets reduced to *k*<sup>0</sup> ¼ ð Þ ð Þ *hk <sup>N</sup>=<sup>k</sup>* where *g h*, *<sup>N</sup> k* � � <sup>¼</sup> 1. The number of such multiplicative factors *<sup>h</sup>* is *<sup>Φ</sup> <sup>N</sup> k* � �, and these factors are the totatives of *<sup>N</sup> <sup>k</sup>* Hence, the number of frequency indices that are related by complete redundancy to a frequency index *k* is given by *Φ <sup>N</sup> k* � � and they are obtained by *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> ð Þ *hk <sup>N</sup>* where *g h*, *<sup>N</sup> k* � � <sup>¼</sup> 1.

From number theory [17],

$$\sum\_{d \mid N \mid N} \Phi(d) = N, \quad \text{if } N \ge 1,\tag{47}$$

#### *A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

Using Eq. (43) and gcd property,

From Eqs. (40) and (44),

*Number Theory and Its Applications*

*<sup>g</sup>*ð Þ¼ 6, 8 2, *<sup>Y</sup>*ð Þ <sup>0</sup>

<sup>2</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup>

*g hk* ð Þ<sup>1</sup> *<sup>N</sup>*, *<sup>N</sup>* � � <sup>¼</sup> *g hk* ð Þ¼ <sup>1</sup> � *Nq*, *<sup>N</sup> g hk* ð Þ¼ 1, *<sup>N</sup> <sup>k</sup>* (44)

From Eqs. (41) and (45), and using Theorem 4, it can be concluded that frequency indices *k*<sup>1</sup> and *k*<sup>2</sup> are completely redundant. Hence, the theorem is proved.

> <sup>5</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup> 7

> > <sup>5</sup> ¼ �*Y*ð Þ<sup>3</sup>

<sup>5</sup> ¼ �*Y*ð Þ<sup>2</sup>

<sup>5</sup> ¼ �*Y*ð Þ<sup>1</sup>

<sup>7</sup> ,

7 *:*

<sup>7</sup> and,

<sup>3</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup>

<sup>3</sup> ¼ �*Y*ð Þ<sup>1</sup>

<sup>3</sup> ¼ �*Y*ð Þ<sup>3</sup>

among the MRT coefficients of these frequencies. Similarly, since *g*ð Þ¼ 2, 8

½ � *hk <sup>N</sup>* ¼ *h* þ

*N k* � �

� � � �

Eq. (46) implies that the set of *k*<sup>0</sup> that is generated from divisor *k* is unique only

*k* � � <sup>¼</sup> 1.

remaining sets of the same length. Hence, the problem now gets reduced to *k*<sup>0</sup> ¼

are related by complete redundancy to a frequency index *k* is given by *Φ <sup>N</sup>*

X *dVN* *k*

� � <sup>¼</sup> 1. The number of such multiplicative factors *<sup>h</sup>* is *<sup>Φ</sup> <sup>N</sup>*

*N*

*<sup>k</sup>* � <sup>1</sup> � �, and repeats thereafter for the

*<sup>k</sup>* Hence, the number of frequency indices that

*Φ*ð Þ¼ *d N*, if *N* ≥1, (47)

(46)

*k* � �,

*k* � � and

<sup>2</sup> ¼ �*Y*ð Þ<sup>2</sup>

<sup>3</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ<sup>2</sup>

Since *k* ¼ 1, 3, 5 and 7 share the same gcd of 1 w.r.t. *N*, there is redundancy

6 . Theorem 5 shows that we can group frequencies based on their gcd w.r.t. *N*. AII nondivisor frequency indices are related to divisor frequency indices through multiplication factors *h* such that *g h*ð Þ¼ , *N* 1. The count of the possible factors of multiplication involved in complete redundancy is Euler's totient function *Φ*ð Þ *N* , which specifies the count of positive integers smaller than or equal to *N* that are coprime to *N*, 1 being considered co-prime to every other integer. Given 1-D MRT coefficients having frequency *k* ¼ 1, there would be *Φ*ð Þ� *N* 1 other frequencies whose MRT coefficients can be obtained from this MRT coefficient. Combined with *k* ¼ 1, these *Φ*ð Þ *N* frequencies thus comprise the group of frequencies that satisfy *g h*ð Þ¼ , *N* 1. There are other sets of frequency indices that have a common gcds w.r.t. *N*.; and each group corresponds to a some divisor of *N*. There are *Φ*ð Þ *N* possible factors of multiplication that produce other members of the group of frequency indices associated with *k*. From Theorem 4, the equation for complete

For example, if *N* ¼ 8, the following relations exist:

*Y*ð Þ <sup>0</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ <sup>0</sup>

*Y*ð Þ<sup>1</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ<sup>3</sup>

*Y*ð Þ<sup>2</sup>

*Y*ð Þ<sup>3</sup> <sup>1</sup> <sup>¼</sup> *<sup>Y</sup>*ð Þ<sup>1</sup>

<sup>6</sup> , and *<sup>Y</sup>*ð Þ<sup>2</sup>

redundancy is *k*<sup>0</sup> ¼ ð Þ *hk <sup>N</sup>*, where *g h*ð Þ¼ , *N* 1. Also,

for multiplicative factors in the set 0, *<sup>N</sup>*

and these factors are the totatives of *<sup>N</sup>*

From number theory [17],

*k*

they are obtained by *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> ð Þ *hk <sup>N</sup>* where *g h*, *<sup>N</sup>*

ð Þ ð Þ *hk <sup>N</sup>=<sup>k</sup>* where *g h*, *<sup>N</sup>*

**14**

<sup>1</sup> ¼ �*Y*ð Þ<sup>2</sup>

*k*<sup>2</sup> ¼ ½½ �� *hk*<sup>1</sup> *<sup>N</sup>* (45)

From Eq. (47), the sum of the terms *Φ <sup>N</sup> k* � � over all divisors of *N* is given by *N*, since P *<sup>d</sup>*∣*NΦ*ð Þ¼ *<sup>d</sup>* <sup>P</sup> *<sup>d</sup>*∣*NΦ <sup>N</sup> d* � �. Hence, all the *<sup>N</sup>* frequency indices *<sup>k</sup>* <sup>¼</sup> ½ � 0, *<sup>N</sup>* � <sup>1</sup> have been mapped.

Thus, all the frequency indices of 1-D MRT can be classified on the basis of their gcd w.r.t. *N*.

#### **3.3 Theorem 6: Mapping between phase indices**

One-to-one mapping exists between phases of 1-D MRT coefficients having frequencies *k* and *k*<sup>0</sup> connected through complete redundancy.

**Proof.** For two MRT coefficients having frequencies *k* and *k*<sup>0</sup> connected by complete redundancy, the number of phases corresponding to each frequency is equal since *g k*ð Þ¼ , *N g k*<sup>0</sup> , *N* � �, and the number of phase indices is given by *<sup>N</sup> g k*ð Þ , *<sup>N</sup>* . The phases lie in the range 0, ½ � *M* � 1 . From Theorem 4 on complete redundancy, the relation between phases is given by *p*<sup>0</sup> ¼ ½½ �� *hp <sup>N</sup>*, *g h*ð Þ¼ , *N* 1. Using a theorem on the reduced residue systems, on multiplication with *h*, the resulting group of phase indices *p*<sup>0</sup> too have the same elements as the original group. Multiplication of phases in 0, ½ � *M* � 1 by *h* and then computing modulo w.r.t. *M* produces the same group, but with the order possibly altered. Thus, one-to-one mapping exists between the phases of 1-D MRT coefficients having frequencies *k* and *k*<sup>0</sup> connected through complete redundancy.

To look at an example, when *N* ¼ 8,

$$\begin{aligned} Y\_1^{(0)} &= Y\_3^{(0)}, \\ Y\_1^{(1)} &= Y\_3^{(3)} \\ Y\_1^{(2)} &= -Y\_3^{(2)}, \text{and}, \\ Y\_1^{(3)} &= Y\_3^{(1)}. \end{aligned}$$

Since *k* ¼ 1 and *k* ¼ 3 are redundant through the co-prime *h* ¼ 3, the set of phase indices of *k* ¼ 1, *p* ¼ ½ � 0, 1, 2, 3 , when subjected to the operation *p*<sup>0</sup> ¼ ½ � *hp <sup>N</sup>* would result in*p*<sup>0</sup> ¼ ½ � 0, 3, 6, 1 , which reduces to *p*<sup>0</sup> ¼ ½ � 0, 3, 2, 1 after the condition ½ �� ½*hp <sup>N</sup>* < *M* is checked and relevant sign change. Hence, *p* ¼ ½ � 0, 1, 2, 3 maps to *p*<sup>0</sup> ¼ ½ � 0, 3, 2, 1 .

#### **3.4 Derived redundancy**

For *N* ¼ 6,

$$Y\_3^{(0)} = Y\_1^{(0)} - Y\_1^{(1)} + Y\_1^{(2)}$$

Certain MRT coefficients can be obtained using the summation of certain other unique MRT coefficients. This phenomenon is also a kind of redundancy, but it cannot be considered as complete redundancy. This phenomenon is referred to as derived redundancy. An MRT coefficient is considered a derived MRT coefficient if we can be obtain it by combining other MRT coefficients.

Let *Y*ð Þ *<sup>p</sup> <sup>k</sup>*<sup>0</sup> and *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup> <sup>k</sup>* be the two MRT coefficients. The congruence relations for *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup> k* are ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa* and ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa* <sup>þ</sup> *<sup>M</sup>*. The congruence relations for *<sup>Y</sup>*ð Þ *<sup>p</sup> <sup>k</sup>*<sup>0</sup> are *nk*<sup>0</sup> � � � *<sup>N</sup>* ¼ *p* and *nk*<sup>0</sup> � � � � *<sup>N</sup>* ¼ *p* þ *M*. Assume that a relation *k*<sup>0</sup> ¼ *dk* exists between *k* and *k*<sup>0</sup> .

#### *Number Theory and Its Applications*


the absolutely unique divisor set {1, 2}. It is not possible to express absolutely unique divisors as *k*<sup>0</sup> ¼ *dk*, for *d* an odd integer. Considering the divisor set of *N*, this condition is satisfied only by divisors that are powers of 2, as observed in the

So, the unique divisor set for *N* contains only divisors which are powers of 2. MRT coefficients formed by unique divisor frequencies are referred to as unique MRT (UMRT) coefficients. Thus, 1-D UMRT is the set of all MRT coefficients having

In total, there are *MN* MRT coefficients given a signal of size *N*. Now arises the

If *N* is a power of 2, the frequencies that form unique coefficients are themselves powers of 2. Since, *k* ¼ 0 ¼ ½½ �� *N <sup>N</sup>* and *N* a power of 2, then *k* ¼ 0 produces a unique coefficient. The first frequency index is *k* ¼ 1, and followed by *k* ¼ 2, 4, …, *N*. The number of unique coefficients produced by each frequency index *k* is given by the number of allowable phase indices for each frequency index *k*. For an MRT coeffi-

*<sup>k</sup>* to exist, *p* should be divisible by *k*. When *k* ¼ *N*, this condition has only

*M* 2*t*

> 1 2*t*

log X<sup>2</sup>*<sup>M</sup> t*¼0

<sup>1</sup> � <sup>1</sup> 2 � � log ð Þ <sup>2</sup>*M*þ<sup>1</sup> 1 2

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>N</sup>* <sup>1</sup> � <sup>2</sup>� log ð Þ <sup>2</sup>*M*þ<sup>1</sup> � �

*N* � �

, then given any *k* and odd-valued *d* except *d* ¼ 1, *k*<sup>0</sup> 6¼ *dk*.

*<sup>k</sup>* coefficients each.

be the

*<sup>k</sup>* MRT

, *d* being an odd

<sup>2</sup> . There are *<sup>M</sup>*

, *d* being odd.

<sup>2</sup> cannot be integer-valued. The number of

*<sup>k</sup>*<sup>0</sup> MRT coefficients in the case that *k*<sup>0</sup>

above example. If *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>a</sup>*

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

cient *Y*ð Þ *<sup>p</sup>*

frequencies that are powers of 2.

**4.1 Number of unique coefficients**

question of the exact number of UMRT coefficients.

one solution for *<sup>p</sup>*, *<sup>p</sup>* <sup>¼</sup> 0. All other frequency indices have *<sup>M</sup>*

*Tot* ¼ 1 þ

¼ 1 þ *M*

¼ 1 þ *M*

¼ *N*

highest power of 2 frequency and a divisor of *N*. Hence *N* ¼ *dk*<sup>0</sup>

*<sup>k</sup>*<sup>0</sup> and thus there exist *<sup>N</sup>*

smaller than *<sup>k</sup>*<sup>0</sup> will be divisors of *<sup>M</sup>*. These are *<sup>k</sup>* <sup>¼</sup> 1, 2, 4, …, *<sup>k</sup>*<sup>0</sup>

*k* ¼ *N*, since ½ �� ½*N <sup>N</sup>* ¼ 0. Frequency *N* relates to *k*<sup>0</sup> through *N* ¼ *dk*<sup>0</sup>

possible for *k*<sup>0</sup> to be a divisor for *M* as *<sup>d</sup>*

valid phase indices is *<sup>N</sup>*

**17**

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>N</sup>* <sup>1</sup> � <sup>1</sup>

Hence, the number of UMRT coefficients, when *N* is a power of 2, is *N*. If *N* is not a power of 2, *k* ¼ 0 does not produce UMRT coefficients. Let *k*<sup>0</sup>

integer; if *d* has an even value, then it would violate the assumption that *k*<sup>0</sup> is the highest power of 2 frequency and a divisor of *N*. Since *d* has an odd value, it is not

does not divide *M*. AII the numbers that are powers of 2 and divide *N* and are

coefficients produced by these frequencies. When *N* is not a power of 2, frequency *k* ¼ 0 can be obtained by derived redundancy. A frequency *k* ¼0 is equivalent to

log X<sup>2</sup>*<sup>M</sup> t*¼0

Hence, the total number of UMRT coefficients is given by

**Table 1.**

*Complete redundancy and derived redundancy relations for N = 6.*


**Table 2.** *Complete redundancy and derived redundancy relations for N = 24.*

Hence, ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* and ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>M</sup>:* ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* may be written as ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup> d* if *g d*ð Þ¼ , *<sup>N</sup> <sup>d</sup>*, and *<sup>d</sup>*∣*p*. In other words, *<sup>d</sup>* should be a divisor of *<sup>N</sup>*. From ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup> <sup>d</sup>* and ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa*, *pa* <sup>¼</sup> *<sup>p</sup> <sup>d</sup>*. Multiplying both sides of ½ �� ½*nk <sup>N</sup>* ¼ *pa* þ *M by d*, ½ �� ½*dnk <sup>N</sup>* ¼ *dpa* þ *dM*. If *d* is odd, this can be written as ½ �� ½*dnk <sup>N</sup>* ¼ *dpa* þ *M*, which becomes *nk*<sup>0</sup> *<sup>N</sup>* ¼ *p* þ *M*. Hence, provided there exists an odd-valued divisor *d* of *N*, such that *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> *dk*, then there is derived redundancy between and *<sup>Y</sup>*ð Þ *<sup>p</sup> <sup>k</sup>*<sup>0</sup> and *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup> <sup>k</sup>* . Hence, we can conclude that there cannot be derived redundancy when *N* is a power of 2 since *N* has no divisors that are odd. For even *N* not a power of 2, derived redundancy exists since *N* then has divisors that are odd. Details of derived redundancies are shown in **Tables 1** and **2** for *N* ¼ 6 and *N* ¼ 24, respectively.

#### **4. Unique MRT**

Based on the two types of redundancy presented above, MRT coefficients can be considered to be unique or relatively unique. It is impossible to obtain MRT coefficients of divisor frequencies from MRT coefficients of other divisor frequencies by complete redundancy. These can hence be named as unique MRT coefficients. If divisors have a relationship to other divisors by multiplication using a divisor that is odd-valued, derived redundancy is exhibited by them. They are thus considered only relatively unique. For *N* ¼ 6, the set of divisor frequencies is {1, 2, 3, 6}. In this set, 3 and 6 are relatively unique divisor frequencies. If we remove these, we obtain

#### *A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

the absolutely unique divisor set {1, 2}. It is not possible to express absolutely unique divisors as *k*<sup>0</sup> ¼ *dk*, for *d* an odd integer. Considering the divisor set of *N*, this condition is satisfied only by divisors that are powers of 2, as observed in the above example. If *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>a</sup>* , then given any *k* and odd-valued *d* except *d* ¼ 1, *k*<sup>0</sup> 6¼ *dk*. So, the unique divisor set for *N* contains only divisors which are powers of 2. MRT coefficients formed by unique divisor frequencies are referred to as unique MRT (UMRT) coefficients. Thus, 1-D UMRT is the set of all MRT coefficients having frequencies that are powers of 2.

#### **4.1 Number of unique coefficients**

In total, there are *MN* MRT coefficients given a signal of size *N*. Now arises the question of the exact number of UMRT coefficients.

If *N* is a power of 2, the frequencies that form unique coefficients are themselves powers of 2. Since, *k* ¼ 0 ¼ ½½ �� *N <sup>N</sup>* and *N* a power of 2, then *k* ¼ 0 produces a unique coefficient. The first frequency index is *k* ¼ 1, and followed by *k* ¼ 2, 4, …, *N*. The number of unique coefficients produced by each frequency index *k* is given by the number of allowable phase indices for each frequency index *k*. For an MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* to exist, *p* should be divisible by *k*. When *k* ¼ *N*, this condition has only one solution for *<sup>p</sup>*, *<sup>p</sup>* <sup>¼</sup> 0. All other frequency indices have *<sup>M</sup> <sup>k</sup>* coefficients each. Hence, the total number of UMRT coefficients is given by

$$\begin{aligned} Tot &= \mathbf{1} + \sum\_{t=0}^{\log\_2 M} \frac{M}{2^t} \\ &= \mathbf{1} + M \sum\_{t=0}^{\log\_2 M} \frac{\mathbf{1}}{2^t} \\ &= \mathbf{1} + M \frac{\mathbf{1} - \left(\frac{1}{2}\right)^{\left(\log\_2 M + 1\right)}}{2} \\ &= \mathbf{1} + N \left(\mathbf{1} - 2^{-\left(\log\_2 M + 1\right)}\right) \\ &= \mathbf{1} + N \left(\mathbf{1} - \frac{\mathbf{1}}{N}\right) \\ &= N \end{aligned}$$

Hence, the number of UMRT coefficients, when *N* is a power of 2, is *N*.

If *N* is not a power of 2, *k* ¼ 0 does not produce UMRT coefficients. Let *k*<sup>0</sup> be the highest power of 2 frequency and a divisor of *N*. Hence *N* ¼ *dk*<sup>0</sup> , *d* being an odd integer; if *d* has an even value, then it would violate the assumption that *k*<sup>0</sup> is the highest power of 2 frequency and a divisor of *N*. Since *d* has an odd value, it is not possible for *k*<sup>0</sup> to be a divisor for *M* as *<sup>d</sup>* <sup>2</sup> cannot be integer-valued. The number of valid phase indices is *<sup>N</sup> <sup>k</sup>*<sup>0</sup> and thus there exist *<sup>N</sup> <sup>k</sup>*<sup>0</sup> MRT coefficients in the case that *k*<sup>0</sup> does not divide *M*. AII the numbers that are powers of 2 and divide *N* and are smaller than *<sup>k</sup>*<sup>0</sup> will be divisors of *<sup>M</sup>*. These are *<sup>k</sup>* <sup>¼</sup> 1, 2, 4, …, *<sup>k</sup>*<sup>0</sup> <sup>2</sup> . There are *<sup>M</sup> <sup>k</sup>* MRT coefficients produced by these frequencies. When *N* is not a power of 2, frequency *k* ¼ 0 can be obtained by derived redundancy. A frequency *k* ¼0 is equivalent to *k* ¼ *N*, since ½ �� ½*N <sup>N</sup>* ¼ 0. Frequency *N* relates to *k*<sup>0</sup> through *N* ¼ *dk*<sup>0</sup> , *d* being odd.

Hence, ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* and ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* <sup>þ</sup> *<sup>M</sup>:* ½ �� <sup>½</sup>*dnk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>* may be written as ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>*

*N* **= 6 Complete redundancy (5) Derived redundancy (3)**

*N* **= 24 Co-primes Odd divisors**

1 5 7 11 13 17 19 23 3 2 10 14 22 6

4 20 12

8 16 24

5 7 11 13 17 19 23 3

15 3 24 6

*Complete redundancy and derived redundancy relations for N = 6.*

3 15 21 9

*Number Theory and Its Applications*

6 18

*<sup>d</sup>*. Multiplying both sides of ½ �� ½*nk <sup>N</sup>* ¼ *pa* þ *M by d*, ½ �� ½*dnk <sup>N</sup>* ¼

*g d*ð Þ¼ , *<sup>N</sup> <sup>d</sup>*, and *<sup>d</sup>*∣*p*. In other words, *<sup>d</sup>* should be a divisor of *<sup>N</sup>*. From ½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>*

*dpa* þ *dM*. If *d* is odd, this can be written as ½ �� ½*dnk <sup>N</sup>* ¼ *dpa* þ *M*, which becomes

we can conclude that there cannot be derived redundancy when *N* is a power of 2

redundancy exists since *N* then has divisors that are odd. Details of derived redun-

Based on the two types of redundancy presented above, MRT coefficients can be considered to be unique or relatively unique. It is impossible to obtain MRT coefficients of divisor frequencies from MRT coefficients of other divisor frequencies by complete redundancy. These can hence be named as unique MRT coefficients. If divisors have a relationship to other divisors by multiplication using a divisor that is odd-valued, derived redundancy is exhibited by them. They are thus considered only relatively unique. For *N* ¼ 6, the set of divisor frequencies is {1, 2, 3, 6}. In this set, 3 and 6 are relatively unique divisor frequencies. If we remove these, we obtain

since *N* has no divisors that are odd. For even *N* not a power of 2, derived

dancies are shown in **Tables 1** and **2** for *N* ¼ 6 and *N* ¼ 24, respectively.

that *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> *dk*, then there is derived redundancy between and *<sup>Y</sup>*ð Þ *<sup>p</sup>*

*Complete redundancy and derived redundancy relations for N = 24.*

*<sup>N</sup>* ¼ *p* þ *M*. Hence, provided there exists an odd-valued divisor *d* of *N*, such

½ �� <sup>½</sup>*nk <sup>N</sup>* <sup>¼</sup> *pa*, *pa* <sup>¼</sup> *<sup>p</sup>*

**4. Unique MRT**

*nk*<sup>0</sup>

**16**

12 24

**Table 2.**

3 6

**Table 1.**

*d* if

*<sup>d</sup>* and

*<sup>k</sup>* . Hence,

*<sup>k</sup>*<sup>0</sup> and *<sup>Y</sup> <sup>p</sup>*ð Þ*<sup>a</sup>*

This can be recalled to be sufficient for derived redundancy to occur. Thus, in case if *N* is not a power of 2, *k* ¼ 0 does not form an absolutely unique MRT coefficient. Hence, the total number of UMRT coefficients is

$$\begin{aligned} Tot &= 1 + \sum\_{t=0}^{\log\_2 M} \frac{M}{2^t} \\ &= 1 + M \frac{1 - \left(\frac{1}{2}\right)^{\left(\log\_2 M + 1\right)}}{\frac{1}{2}} \\ &= \frac{N}{k'} + N \left(1 - \frac{2}{2k'}\right) \\ &= N \end{aligned}$$

We thus conclude that *N* UMRT coefficients are needed to represent a 1-D signal of length *N*, whether *N* is a power of 2 or not, and these coefficients are produced by frequencies that are divisors of *N* and powers of 2, beginning with *k* ¼ 1.

The 1-D UMRT coefficients can be computed as below:

(i) *N* a power of 2

$$Y\_0^{(0)} = \sum\_{n=0}^{N-1} \mathfrak{x}\_n \tag{48}$$

**5. Inverse MRT**

coefficient *Y*ð Þ *<sup>p</sup>*

data *xn*.

*Y*ð Þ <sup>0</sup>

**19**

coefficients *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

example, for *<sup>N</sup>* <sup>¼</sup> 8, *<sup>Y</sup>*ð Þ <sup>0</sup>

A UMRT coefficient *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

corresponding to *<sup>k</sup>* <sup>¼</sup> 1, given *byY*ð Þ <sup>0</sup>

UMRT coefficient *Y*ð Þ <sup>0</sup>

**5.1 Theorem 7: N a power of 2**

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

*xn* <sup>¼</sup> <sup>1</sup> *N Y*ð Þ <sup>0</sup> 0 þ

ð Þ *nk <sup>N</sup>* <sup>¼</sup> 0, <sup>∀</sup>*n*, when *<sup>k</sup>* <sup>¼</sup> 0. In Eq. (53), *<sup>Y</sup>*ð Þ <sup>0</sup>

a sum of the individual factors of multiplication.

Given the UMRT of a 1-D signal of size *N*, *N* being a power of 2, the 1-D signal

**Proof.** The data element that needs to be recovered from the UMRT is given by

*xn* that is a part of these coefficients have the corresponding factors of multiplication. The resultant multiplication factor *f* for *xn* due to the summation is obtained as

> 1 2*<sup>t</sup>*þ<sup>1</sup>

2�ð Þ *<sup>t</sup>*þ<sup>1</sup>

<sup>1</sup> � <sup>2</sup>� log ð Þ <sup>2</sup>*M*þ<sup>1</sup> � � 1 2

log X<sup>2</sup>*<sup>M</sup> t*¼0

log X<sup>2</sup>*<sup>M</sup> t*¼0

*<sup>N</sup>* <sup>þ</sup> <sup>1</sup> � <sup>1</sup>

*N*

Hence, as a result of the summation, one of the components of the result is the

transform formula to be correct, these other terms that occur in the various UMRT

*k* be the smallest frequency at which any other data elements co-occur with *xn*. *Y*ð Þ <sup>0</sup>

can be excluded as all data elements co-occur in it with a positive sign. Leaving out

<sup>0</sup> , another element shows co-occurrence with *xn* for the first time for *k* ¼ 1. For

seen that *x*<sup>4</sup> occurs with an opposite sign along with *x*<sup>0</sup> in the MRT coefficient

so both positive and negative data sets contain only one element each. Using

<sup>2</sup>*<sup>a</sup>* contains other terms besides *xn*. For the inverse

<sup>1</sup> ¼ *x*<sup>0</sup> � *x*4. Hence, if *x*<sup>0</sup> is the element to be obtained, it is

<sup>1</sup> . From Eq. (12), when *k* ¼ 1, *g k*ð Þ¼ , *N* 1 and

<sup>2</sup>*<sup>a</sup>* need to get canceled off. It can be proved that they vanish. Let

*<sup>k</sup>* that contains *xn* is given by nk ð Þ *nk <sup>N</sup>* ¼ *p*. Thus for a frequency that

<sup>0</sup> contains all the elements of the data including *xn* since

<sup>2</sup>*<sup>t</sup>* , 0≤*n* ≤ *N* � 1 (53)

<sup>0</sup> has a factor of multiplication <sup>1</sup>

<sup>2</sup>*<sup>a</sup>* . The

<sup>2</sup>*t*þ1. Consequently, the

*<sup>N</sup>*, and the

0

1 <sup>2</sup>*t*þ<sup>1</sup> *<sup>Y</sup>* <sup>2</sup>*<sup>t</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

*xn*. For any frequency index *k*, the value of the phase index *p* of the UMRT

is a power of 2, *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>*, the UMRT coefficient that contains *xn* is *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

remaining UMRT coefficients have a factor of multiplication <sup>1</sup>

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *N* þ

> ¼ 1 *N* þ

¼ 1 *N* þ 1 2

¼ 1

¼ 1

can be reconstructed from its UMRT by the following formula

log X<sup>2</sup>*<sup>M</sup> t*¼0

$$\begin{aligned} Y\_k^{(p)} &= \sum\_{j=0}^{k-1} \left( \underline{\chi\_{\frac{(j+1)p}{k}}} - \underline{\chi\_{\frac{(j+1)N+2p}{2k}}} \right) \\\\ k &= 2^t, 0 \le t \le \log\_2 M, p = tk, 0 \le t \le \frac{M}{k} - 1 \end{aligned} \tag{49}$$

(ii) *N* not a power of 2

$$Y\_k^{(p)} = \sum\_{j=0}^{k-1} \left( \mathbf{x}\_{\frac{2k+p}{k}} - \mathbf{x}\_{\frac{(2j+1)k+2p}{2k}} \right) \tag{50}$$

$$k = 2^t, 0 \le t \le \log\_2 M, p = tk, \quad 0 \le t \le \frac{M}{k} - 1$$

$$Y\_{k'}^{(p)} = \sum\_{j=0}^{k'-1} \left( \mathbf{x}\_{\frac{2k+p}{k}} \right)$$

$$p = tk', \quad 0 \le t \le \frac{M}{k'} - \frac{1}{2}$$

$$Y\_{k'}^{(p)} = -\sum\_{j=0}^{k'-1} \left( \mathbf{x}\_{\frac{2k+p+M}{k}} \right) \tag{52}$$

$$p = tk' + \frac{k'}{2}, \quad 0 \le t \le \frac{M}{k'} - \frac{3}{2}$$

where *k*<sup>0</sup> is the highest frequency index that is a power of 2 and also a divisor of *N*.

#### **5. Inverse MRT**

This can be recalled to be sufficient for derived redundancy to occur. Thus, in case if *N* is not a power of 2, *k* ¼ 0 does not form an absolutely unique MRT coefficient.

> *M* 2*t*

<sup>1</sup> � <sup>1</sup> 2 � � log ð Þ <sup>2</sup>*M*þ<sup>1</sup> 1 2

*<sup>k</sup>*<sup>0</sup> <sup>þ</sup> *<sup>N</sup>* <sup>1</sup> � <sup>2</sup>

We thus conclude that *N* UMRT coefficients are needed to represent a 1-D signal of length *N*, whether *N* is a power of 2 or not, and these coefficients are produced by frequencies that are divisors of *N* and powers of 2, beginning with *k* ¼ 1.

*n*¼0

� *x*ð Þ <sup>2</sup>*j*þ<sup>1</sup> *<sup>N</sup>*þ2*<sup>p</sup>* 2*k*

, 0 ≤*t* ≤ log <sup>2</sup> *M*, *p* ¼ *tk*, 0 ≤*t*≤

� *x*ð Þ <sup>2</sup>*j*þ<sup>1</sup> *<sup>N</sup>*þ2*<sup>p</sup>* 2*k*

, 0≤ *t*≤ log <sup>2</sup> *M*, *p* ¼ *tk*, 0≤*t*≤

*xjN*þ*<sup>p</sup> k* � �

> *M <sup>k</sup>*<sup>0</sup> � <sup>1</sup> 2

*xjN*þ*p*þ*<sup>M</sup> k*0 � �

> *M <sup>k</sup>*<sup>0</sup> � <sup>3</sup> 2

is the highest frequency index that is a power of 2 and also a divisor of *N*.

*j*¼0

*<sup>k</sup>*<sup>0</sup> ¼ �<sup>X</sup> *k*0 �1

, 0≤ *t*≤

*j*¼0

<sup>2</sup> , 0≤*t*<sup>≤</sup>

� �

� �

*xn* (48)

(49)

(50)

(51)

(52)

*M <sup>k</sup>* � <sup>1</sup>

> *M <sup>k</sup>* � <sup>1</sup>

2*k*<sup>0</sup> � �

log X<sup>2</sup>*<sup>M</sup> t*¼0

Hence, the total number of UMRT coefficients is

*Number Theory and Its Applications*

*Tot* ¼ 1 þ

¼ 1 þ *M*

¼ *N*

¼ *N*

*Y*ð Þ <sup>0</sup> <sup>0</sup> <sup>¼</sup> <sup>X</sup> *N*�1

*xjN*þ*<sup>p</sup> k*

*xjN*þ*<sup>p</sup> k*

*Y*ð Þ *<sup>p</sup> <sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>X</sup> *k*0 �1

*p* ¼ *tk*<sup>0</sup>

*Y*ð Þ *<sup>p</sup>*

þ *k*0

*p* ¼ *tk*<sup>0</sup>

The 1-D UMRT coefficients can be computed as below:

*j*¼0

*j*¼0

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* <sup>¼</sup> <sup>X</sup> *k*�1

*<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>t</sup>*

*Y*ð Þ *<sup>p</sup> <sup>k</sup>* <sup>¼</sup> <sup>X</sup> *k*�1

*<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>t</sup>*

(i) *N* a power of 2

(ii) *N* not a power of 2

where *k*<sup>0</sup>

**18**

#### **5.1 Theorem 7: N a power of 2**

Given the UMRT of a 1-D signal of size *N*, *N* being a power of 2, the 1-D signal can be reconstructed from its UMRT by the following formula

$$\mathbf{x}\_{\mathfrak{n}} = \frac{\mathbf{1}}{N} Y\_0^{(0)} + \sum\_{t=0}^{\log\_2 M} \frac{\mathbf{1}}{\mathfrak{2}^{t+1}} Y\_{\mathfrak{2}^t}^{\left( (\mathfrak{X}n)\_N \right)}, \ \mathbf{0} \le n \le N - \mathbf{1} \tag{53}$$

**Proof.** The data element that needs to be recovered from the UMRT is given by *xn*. For any frequency index *k*, the value of the phase index *p* of the UMRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>* that contains *xn* is given by nk ð Þ *nk <sup>N</sup>* ¼ *p*. Thus for a frequency that is a power of 2, *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>*, the UMRT coefficient that contains *xn* is *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>* <sup>2</sup>*<sup>a</sup>* . The UMRT coefficient *Y*ð Þ <sup>0</sup> <sup>0</sup> contains all the elements of the data including *xn* since ð Þ *nk <sup>N</sup>* <sup>¼</sup> 0, <sup>∀</sup>*n*, when *<sup>k</sup>* <sup>¼</sup> 0. In Eq. (53), *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>0</sup> has a factor of multiplication <sup>1</sup> *<sup>N</sup>*, and the remaining UMRT coefficients have a factor of multiplication <sup>1</sup> <sup>2</sup>*t*þ1. Consequently, the *xn* that is a part of these coefficients have the corresponding factors of multiplication. The resultant multiplication factor *f* for *xn* due to the summation is obtained as a sum of the individual factors of multiplication.

$$\begin{aligned} f &= \frac{1}{N} + \sum\_{t=0}^{\log\_2 M} \frac{1}{2^{t+1}} \\ &= \frac{1}{N} + \sum\_{t=0}^{\log\_2 M} 2^{-(t+1)} \\ &= \frac{1}{N} + \frac{1}{2} \frac{\left(1 - 2^{-\left(\log\_2 M + 1\right)}\right)}{\frac{1}{2}} \\ &= \frac{1}{N} + 1 - \frac{1}{N} \\ &= 1 \end{aligned}$$

Hence, as a result of the summation, one of the components of the result is the data *xn*.

A UMRT coefficient *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>* <sup>2</sup>*<sup>a</sup>* contains other terms besides *xn*. For the inverse transform formula to be correct, these other terms that occur in the various UMRT coefficients *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>* <sup>2</sup>*<sup>a</sup>* need to get canceled off. It can be proved that they vanish. Let *k* be the smallest frequency at which any other data elements co-occur with *xn*. *Y*ð Þ <sup>0</sup> 0 can be excluded as all data elements co-occur in it with a positive sign. Leaving out *Y*ð Þ <sup>0</sup> <sup>0</sup> , another element shows co-occurrence with *xn* for the first time for *k* ¼ 1. For example, for *<sup>N</sup>* <sup>¼</sup> 8, *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>1</sup> ¼ *x*<sup>0</sup> � *x*4. Hence, if *x*<sup>0</sup> is the element to be obtained, it is seen that *x*<sup>4</sup> occurs with an opposite sign along with *x*<sup>0</sup> in the MRT coefficient corresponding to *<sup>k</sup>* <sup>¼</sup> 1, given *byY*ð Þ <sup>0</sup> <sup>1</sup> . From Eq. (12), when *k* ¼ 1, *g k*ð Þ¼ , *N* 1 and so both positive and negative data sets contain only one element each. Using

Theorem 3(b), the distance between corresponding elements in the two data groups is *<sup>M</sup> <sup>k</sup>* . Hence, *xn* and *xn*þ*M*, occur having opposite polarity, in any MRT coefficient of frequency *k* ¼ 1. In the same way, for *k* ¼ 2, the data element *xn*þ*M*occurs with positive sign since *Y*ð Þ *<sup>p</sup>* <sup>2</sup> ¼ *xn* � *xn*þ*<sup>N</sup>* <sup>4</sup> þ *xn*þ*<sup>N</sup>* <sup>2</sup> � *xn*þ3*<sup>N</sup>* <sup>4</sup> , given ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*. From (12), the distance between the two successive data elements in a positive or negative group is given by *<sup>N</sup> g k*ð Þ , *<sup>N</sup>* . Since k is a divisor of *<sup>N</sup>*, this reduces to *<sup>N</sup> <sup>k</sup>* . Using Theorem 3 (b), if elements *xn*0, and *xn* co-occur in *Y*ð Þ *<sup>p</sup> <sup>k</sup>* but having different signs, then,

$$n'=n+q\_{odd}\frac{N}{2k}\tag{54}$$

The number of terms in this summation is log <sup>2</sup>ð Þ� *q* 1.

<sup>2</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>f</sup> <sup>a</sup>* <sup>¼</sup> <sup>1</sup>

Hence, the formula for inverse UMRT is proved.

*xn* <sup>¼</sup> <sup>1</sup> *<sup>k</sup>*<sup>0</sup> *Yk*<sup>0</sup>

*nk*<sup>0</sup> ð Þ*<sup>N</sup>* ð Þ

**5.3 General formula for any even N**

signal of size *N*, *N* being any even number.

*xn* <sup>¼</sup> <sup>1</sup>

where *k*<sup>0</sup> is the highest power of 2 divisor of *N*.

<sup>1</sup> � <sup>2</sup>� log <sup>2</sup> ð Þ ð Þ�*<sup>q</sup>* <sup>1</sup> 1 2

*<sup>N</sup>* � <sup>1</sup>

¼ *q* 2*N*

<sup>2</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* � <sup>2</sup>

Thus, all the other data elements *xn*<sup>0</sup> that occur along with *xn* in the various MRT coefficients in the summation of the inverse formula cancel out, leaving behind only

Given the UMRT of a 1-D signal of size *N*, *N* not being a power of 2, the 1-D

*t*¼0

1 <sup>2</sup>*<sup>t</sup>*þ<sup>1</sup> *<sup>Y</sup>* <sup>2</sup>*<sup>t</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

is the highest frequency index that is a power of 2 and also a divisor of *N*.

<sup>2</sup>*t*þ1. The resultant factor *f* that multiplies *xn* as a result of the summa-

signal can be reconstructed from its UMRT by the following formula

*nk*<sup>0</sup> ð Þ*<sup>N</sup>* ð Þ þ log X2*k*0 �1

is multiplied by <sup>1</sup>

only the desired data element *xn*. Hence, the proposed formula is proved.

tion can be proved by using the same method used earlier. Similarly, it can also be shown that data elements other than *xn* cancel out in the summation, leaving behind

Equation (58) can be generalized to be applicable to any even value of *N*. Hence,

the following equation can be used for signal reconstruction from UMRT, for a

þ log X2*k*0 �1

For *N* a power of 2, the UMRT basis matrix can be defined in a new form by combining the frequency index *k* and the phase index *p* of an MRT coefficient into

*H*0ð Þ¼ *m H*0,0ð Þ¼ *m* 1

*t*¼0

1, ½ � *nk <sup>N</sup>* ¼ *p*

1 <sup>2</sup>*<sup>t</sup>*þ<sup>1</sup> *<sup>Y</sup>* <sup>2</sup>*<sup>t</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

*<sup>k</sup>*<sup>0</sup> *<sup>Y</sup> <sup>k</sup>*<sup>0</sup> ð Þ*<sup>N</sup> nk*<sup>0</sup> ð Þ*<sup>N</sup>* ð Þ

**Proof.** *xn* has to be obtained from the UMRT. Given *Y*ð Þ *<sup>p</sup>*

satisfy ð Þ ð Þ *nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>*. For *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>*, the UMRT coefficient *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>*

*q* � 2

<sup>2</sup>*<sup>N</sup>* <sup>¼</sup> *<sup>q</sup>*

*<sup>q</sup>* <sup>¼</sup> *<sup>q</sup>* � <sup>2</sup> 2*N*

<sup>2</sup>*<sup>N</sup>* � <sup>1</sup>

<sup>2</sup>*<sup>k</sup>* <sup>¼</sup> <sup>0</sup>

<sup>2</sup>*<sup>t</sup>* (58)

<sup>2</sup>*<sup>a</sup>* contains *xn*. The

<sup>2</sup>*<sup>t</sup>* (59)

*<sup>k</sup>*<sup>0</sup>, and the other UMRT coefficients are

*<sup>k</sup>* data element *xn* has to

*<sup>f</sup> <sup>a</sup>* <sup>¼</sup> *<sup>q</sup>* 4*N*

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *<sup>N</sup>* � <sup>1</sup>

From Eq. (57),

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

the desired data element *xn*.

where *k*<sup>0</sup>

multiplied by <sup>1</sup>

UMRT coefficient *Yk*<sup>0</sup>

an index *q* as follows:

**21**

**5.2 Theorem 8: N not a power of 2**

where *qodd*is an odd integer. At the next higher frequency *k*<sup>0</sup> ¼ 2*k*, Eq. (54) becomes

$$n' = n + q\_{odd} \frac{N}{k'} \tag{55}$$

From (12), the general form for a data element *xn*<sup>0</sup> of same sign present along with an element *xn* in the MRT coefficient *Y*ð Þ *<sup>p</sup> <sup>k</sup>*<sup>0</sup> is given by (since *k*<sup>0</sup> is a divisor of *N*, *g k*<sup>0</sup> , *<sup>N</sup>* � � <sup>¼</sup> *<sup>k</sup>*<sup>0</sup> Þ

$$n' = n + j\frac{N}{k'} \tag{56}$$

where *j* ¼ 0, 1, 2, 3, …, *k*<sup>0</sup> � 1.

Equation (55) can be seen to be a special case of Eq. (56). For any higher frequency *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*, Eq. (56) holds. Hence, given *<sup>k</sup>* is the lowest frequency at which *xn*<sup>0</sup> and *xn* occur with opposite signs, for all larger frequencies *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*, *xn*<sup>0</sup> and *xn* occur with similar signs. For *<sup>N</sup>* <sup>¼</sup> 8, *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> � *<sup>x</sup>*4, *<sup>Y</sup>*ð Þ <sup>0</sup> <sup>2</sup> ¼ *x*<sup>0</sup> � *x*<sup>2</sup> þ *x*<sup>4</sup> � *x*<sup>6</sup> and *Y*ð Þ <sup>0</sup> <sup>4</sup> ¼ *x*<sup>0</sup> � *x*<sup>1</sup> þ *x*<sup>2</sup> � *x*<sup>3</sup> þ *x*<sup>4</sup> � *x*<sup>5</sup> þ *x*<sup>6</sup> � *x*7. Data elements *x*<sup>4</sup> and *x*<sup>0</sup> occur with opposite signs in *Y*ð Þ <sup>0</sup> <sup>1</sup> and similar signs in higher frequencies, *k* ¼ 2, *k* ¼ 4. Conversely, another conclusion is that elements *xn*<sup>0</sup> and *xn* that co-occur with same signs in a UMRT coefficient having frequency *k*<sup>0</sup> also co-occur with opposite signs in a UMRT coefficient having frequency *<sup>k</sup>* where *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*.

Given *k* is the lowest frequency where *xn*<sup>0</sup> and *xn* co-occur with an opposite sign, the factor of multiplication for *xn*<sup>0</sup> in the inverse transform formula is � <sup>1</sup> <sup>2</sup>*<sup>k</sup>*. In the case of higher frequencies till *M*, the factor of multiplication is <sup>1</sup> <sup>2</sup>*k*<sup>0</sup> , *k*<sup>0</sup> ¼ 2*k*, 4*k*, …*M*. Thus the sum of the series

$$f = \frac{1}{N} - \frac{1}{2k} + \frac{1}{4k} + \frac{1}{8k} + \dots \\ \frac{1}{2M} \tag{57}$$

will provide the value of the multiplication factor *f* associated with element *xn*<sup>0</sup> . Assume *<sup>k</sup>* <sup>¼</sup> *<sup>N</sup> <sup>q</sup>* . First the sum of the following series can be found:

$$\frac{1}{4k} + \frac{1}{8k} + \dots \frac{1}{2M} = f\_{ab}$$

$$f\_{\
u} = \sum\_{j=\log\_2}^{\log\_2 M} \frac{1}{2^{j+1}}$$

The number of terms in this summation is log <sup>2</sup>ð Þ� *q* 1.

$$f\_a = \frac{q}{4N} \frac{1 - 2^{-\left(\log\_2(q) - 1\right)}}{\frac{1}{2}} = \frac{q}{2N} \frac{q - 2}{q} = \frac{q - 2}{2N}$$

From Eq. (57),

Theorem 3(b), the distance between corresponding elements in the two data groups

<sup>4</sup> þ *xn*þ*<sup>N</sup>*

the distance between the two successive data elements in a positive or negative

*g k*ð Þ , *<sup>N</sup>* . Since k is a divisor of *<sup>N</sup>*, this reduces to *<sup>N</sup>*

*n*<sup>0</sup> ¼ *n* þ *qodd*

where *qodd*is an odd integer. At the next higher frequency *k*<sup>0</sup> ¼ 2*k*, Eq. (54)

*n*<sup>0</sup> ¼ *n* þ *qodd*

*n*<sup>0</sup> ¼ *n* þ *j*

Equation (55) can be seen to be a special case of Eq. (56). For any higher frequency *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*, Eq. (56) holds. Hence, given *<sup>k</sup>* is the lowest frequency at which *xn*<sup>0</sup> and *xn* occur with opposite signs, for all larger frequencies *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*, *xn*<sup>0</sup> and *xn*

<sup>4</sup> ¼ *x*<sup>0</sup> � *x*<sup>1</sup> þ *x*<sup>2</sup> � *x*<sup>3</sup> þ *x*<sup>4</sup> � *x*<sup>5</sup> þ *x*<sup>6</sup> � *x*7. Data elements *x*<sup>4</sup> and *x*<sup>0</sup> occur with

versely, another conclusion is that elements *xn*<sup>0</sup> and *xn* that co-occur with same signs in a UMRT coefficient having frequency *k*<sup>0</sup> also co-occur with opposite signs in a

Given *k* is the lowest frequency where *xn*<sup>0</sup> and *xn* co-occur with an opposite sign,

1 4*k* þ

will provide the value of the multiplication factor *f* associated with element *xn*<sup>0</sup> .

*<sup>q</sup>* . First the sum of the following series can be found:

2*N q*

1 <sup>8</sup>*<sup>k</sup>* <sup>þ</sup> … <sup>1</sup>

<sup>2</sup>*<sup>M</sup>* <sup>¼</sup> *<sup>f</sup> <sup>a</sup>*

1 2*<sup>j</sup>*þ<sup>1</sup>

the factor of multiplication for *xn*<sup>0</sup> in the inverse transform formula is � <sup>1</sup>

case of higher frequencies till *M*, the factor of multiplication is <sup>1</sup>

1 4*k* þ

*f <sup>a</sup>* ¼

1 <sup>8</sup>*<sup>k</sup>* <sup>þ</sup> … <sup>1</sup>

> log X<sup>2</sup>*<sup>M</sup>*

*j*¼ log <sup>2</sup>

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *<sup>N</sup>* � <sup>1</sup> 2*k* þ

From (12), the general form for a data element *xn*<sup>0</sup> of same sign present along

<sup>2</sup> ¼ *xn* � *xn*þ*<sup>N</sup>*

(b), if elements *xn*0, and *xn* co-occur in *Y*ð Þ *<sup>p</sup>*

with an element *xn* in the MRT coefficient *Y*ð Þ *<sup>p</sup>*

*<sup>k</sup>* . Hence, *xn* and *xn*þ*M*, occur having opposite polarity, in any MRT coefficient of frequency *k* ¼ 1. In the same way, for *k* ¼ 2, the data element *xn*þ*M*occurs with

<sup>2</sup> � *xn*þ3*<sup>N</sup>*

*N*

*N*

*N*

<sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> � *<sup>x</sup>*4, *<sup>Y</sup>*ð Þ <sup>0</sup>

<sup>1</sup> and similar signs in higher frequencies, *k* ¼ 2, *k* ¼ 4. Con-

<sup>4</sup> , given ð Þ ð Þ *nk <sup>N</sup>* ¼ *p*. From (12),

<sup>2</sup>*<sup>k</sup>* (54)

*<sup>k</sup>*<sup>0</sup> (55)

*<sup>k</sup>*<sup>0</sup> is given by (since *k*<sup>0</sup> is a divisor of

*<sup>k</sup>*<sup>0</sup> (56)

<sup>2</sup> ¼ *x*<sup>0</sup> � *x*<sup>2</sup> þ *x*<sup>4</sup> � *x*<sup>6</sup> and

<sup>2</sup>*<sup>k</sup>*. In the

<sup>2</sup>*k*<sup>0</sup> , *k*<sup>0</sup> ¼ 2*k*, 4*k*, …*M*.

<sup>2</sup>*<sup>M</sup>* (57)

*<sup>k</sup>* but having different signs, then,

*<sup>k</sup>* . Using Theorem 3

is *<sup>M</sup>*

becomes

*N*, *g k*<sup>0</sup>

*Y*ð Þ <sup>0</sup>

**20**

, *<sup>N</sup>* � � <sup>¼</sup> *<sup>k</sup>*<sup>0</sup>

opposite signs in *Y*ð Þ <sup>0</sup>

Thus the sum of the series

Assume *<sup>k</sup>* <sup>¼</sup> *<sup>N</sup>*

Þ

where *j* ¼ 0, 1, 2, 3, …, *k*<sup>0</sup> � 1.

occur with similar signs. For *<sup>N</sup>* <sup>¼</sup> 8, *<sup>Y</sup>*ð Þ <sup>0</sup>

UMRT coefficient having frequency *<sup>k</sup>* where *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>k</sup>*.

positive sign since *Y*ð Þ *<sup>p</sup>*

*Number Theory and Its Applications*

group is given by *<sup>N</sup>*

$$f = \frac{1}{N} - \frac{1}{2k} + f\_a = \frac{1}{N} - \frac{1}{2k} + \frac{q - 2}{2N} = \frac{q}{2N} - \frac{1}{2k} = 0$$

Thus, all the other data elements *xn*<sup>0</sup> that occur along with *xn* in the various MRT coefficients in the summation of the inverse formula cancel out, leaving behind only the desired data element *xn*.

Hence, the formula for inverse UMRT is proved.

#### **5.2 Theorem 8: N not a power of 2**

Given the UMRT of a 1-D signal of size *N*, *N* not being a power of 2, the 1-D signal can be reconstructed from its UMRT by the following formula

$$\infty\_{\mathfrak{n}} = \frac{\mathbf{1}}{k'} Y\_{k'\_{((\omega')\_N)}} + \sum\_{t=0}^{\log\_2 k'-1} \frac{\mathbf{1}}{2^{t+1}} Y\_{2^t}^{\left( (2^t \mathfrak{n})\_N \right)} \tag{58}$$

where *k*<sup>0</sup> is the highest frequency index that is a power of 2 and also a divisor of *N*. **Proof.** *xn* has to be obtained from the UMRT. Given *Y*ð Þ *<sup>p</sup> <sup>k</sup>* data element *xn* has to satisfy ð Þ ð Þ *nk <sup>N</sup>* <sup>¼</sup> *<sup>p</sup>*. For *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>*, the UMRT coefficient *<sup>Y</sup>* <sup>2</sup>*<sup>a</sup>* ð Þ *<sup>n</sup>* ð Þ*<sup>N</sup>* <sup>2</sup>*<sup>a</sup>* contains *xn*. The UMRT coefficient *Yk*<sup>0</sup> *nk*<sup>0</sup> ð Þ*<sup>N</sup>* ð Þ is multiplied by <sup>1</sup> *<sup>k</sup>*<sup>0</sup>, and the other UMRT coefficients are multiplied by <sup>1</sup> <sup>2</sup>*t*þ1. The resultant factor *f* that multiplies *xn* as a result of the summation can be proved by using the same method used earlier. Similarly, it can also be shown that data elements other than *xn* cancel out in the summation, leaving behind only the desired data element *xn*. Hence, the proposed formula is proved.

#### **5.3 General formula for any even N**

Equation (58) can be generalized to be applicable to any even value of *N*. Hence, the following equation can be used for signal reconstruction from UMRT, for a signal of size *N*, *N* being any even number.

$$\mathbf{x}\_n = \frac{\mathbf{1}}{\mathbf{k}'} \mathbf{Y}\_{(\mathbf{k}')\_{N\left((\mathbf{n}')\_N\right)}} + \sum\_{t=0}^{\log\_2 k'-1} \frac{\mathbf{1}}{\mathbf{2}^{t+1}} \mathbf{Y}\_{\mathbf{2}^t}^{\left((\mathbf{2}^t \mathbf{n})\_N\right)} \tag{59}$$

where *k*<sup>0</sup> is the highest power of 2 divisor of *N*.

For *N* a power of 2, the UMRT basis matrix can be defined in a new form by combining the frequency index *k* and the phase index *p* of an MRT coefficient into an index *q* as follows:

$$\begin{aligned} H\_0(m) &= H\_{0,0}(m) = 1 \\ \mathbf{1}, \ [nk]\_N &= p \end{aligned}$$

$$H\_q(m) = H\_{k,p}(m) = \begin{cases} -1, & [nk]\_N = p + \frac{N}{2} \\ \end{cases}$$

$$\begin{array}{rcl} 0, & otherwise \\ \end{array}$$

$$\begin{array}{rcl} q & = 2^k + p & -1 \\ m = 0, 1, ..., N & -1 \\ \end{array}$$

This structure of the UMRT definition has a similar form as that of the Haar transforms [10]. **Table** 3 shows the mapping for *N* ¼ 8.


#### **Table 3.**

*Proposed mapping between 1-D UMRT indices and array indices, for N = 8.*

#### **6. Conclusions and future development**

The MRT is a new representation of signals and involves only real additions. However, the MRT is expansive and redundant. The UMRT removes these features of MRT to give a real, invertible, nonexpansive signal transform for any even values of *N*. The MRT belongs in the same family of transforms as the Haar transform and the Hadamard transform. A number-theoretical foundation has hereby been laid for the 1-D MRT. However, the 2-D version of the MRT also exists. The interconnections between the 1-D version and 2-D version need to be studied further. The real merit of this transform is its utmost simplicity in terms of computational requirements. Simple integer-to-integer transforms will retain their attractiveness in the light of the ongoing switch to Internet of Things (IoT) and edge computing. A potential application of the MRT is its use as a feature vector in various contexts, just like how the Hadamard Transform has been used, as shown in [18–20]. Numerous other applications await the MRT.

#### **Acknowledgements**

The author would like to acknowledge the guidance and contributions of Dr. R. Gopikakumari.

**Author details**

**23**

Rajesh Cherian Roy

*A New Integer-to-Integer Transform*

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

Muthoot Institute of Technology and Science, Ernakulam, Kerala, India

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: rajeshcherian@yahoo.co.in

provided the original work is properly cited.

*A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

*Hq*ð Þ¼ *m Hk*,*p*ð Þ¼ *m*

*Number Theory and Its Applications*

transforms [10]. **Table** 3 shows the mapping for *N* ¼ 8.

*Proposed mapping between 1-D UMRT indices and array indices, for N = 8.*

**6. Conclusions and future development**

Numerous other applications await the MRT.

**Acknowledgements**

**Table 3.**

Dr. R. Gopikakumari.

**22**

(

0, *otherwise*

This structure of the UMRT definition has a similar form as that of the Haar

*k*,*p* 0,0 1,0 1,1 1,2 1,3 2,0 2,2 4,0 *q* 0 12345 6 7

The MRT is a new representation of signals and involves only real additions. However, the MRT is expansive and redundant. The UMRT removes these features of MRT to give a real, invertible, nonexpansive signal transform for any even values of *N*. The MRT belongs in the same family of transforms as the Haar transform and the Hadamard transform. A number-theoretical foundation has hereby been laid for the 1-D MRT. However, the 2-D version of the MRT also exists. The interconnections between the 1-D version and 2-D version need to be studied further. The real merit of this transform is its utmost simplicity in terms of computational requirements. Simple integer-to-integer transforms will retain their attractiveness in the light of the ongoing switch to Internet of Things (IoT) and edge computing. A potential application of the MRT is its use as a feature vector in various contexts, just like how the Hadamard Transform has been used, as shown in [18–20].

The author would like to acknowledge the guidance and contributions of

*<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>p</sup>* � <sup>1</sup> *m* ¼ 0, 1, …, *N* � 1

�1, ½ � *nk <sup>N</sup>* ¼ *p* þ

*N* 2

### **Author details**

Rajesh Cherian Roy Muthoot Institute of Technology and Science, Ernakulam, Kerala, India

\*Address all correspondence to: rajeshcherian@yahoo.co.in

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Ahmed N et al. Discrete cosine transform. IEEE Transactions on Computers. 1974;**C-23**:90-93. DOI: 10.1109/T-C.1974.223784

[2] Bracewell RN. Discrete Hartley transform. Journal of the Optical Society of America. 1983;**73**:1832-1835. DOI: 10.1364/JOSA.73.001832

[3] Ersoy OK. A two-stage representation of DFT and its applications. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1987;**35**:825-831. DOI: 10.1109/TASSP.1987.1165202

[4] Roy RC, Gopikakumari R. A new transform for 2-d signal representation (MRT) and some of its properties. In: Proceedings of the International Conference in Signal Processing and Communication (SPCOM '04). Bangalore: IEEE; 2004. pp. 363-367. DOI: 10.1109/SPCOM.2004.1458423

[5] Lun DPK, Hsung TC, Shen TW. Orthogonal discrete periodic Radon transform. Part I: Theory and realization. Signal Processing. 2003;**83**: 941-955. DOI: 10.1016/S0165-1684(02) 00498-X

[6] Roy RC, Anish Kumar MS, Gopikakumari R. An invertible transform for image representation and its application to image compression. In: Proceedings of the International Symposium in Signal Processing (ISSPA '07). Sharjah: IEEE; 2007. DOI: 10.1109/ ISSPA.2007.4555504

[7] Anish Kumar MS, Roy RC, Gopikakumari R. A new transform coder for gray scale images using 44 MRT. Elsevier AEU—International Journal of Electronics and Communications. 2008;**62–8**:627-630. DOI: 10.1016/j.aeue.2007.04.009

[8] Roy RC, Gopikakumari R. 1-D MRT: A new transform for 1-D signals. In:

Proceedings of the all India Seminar on Innovative Electronics Technology and Futuristic Communication, Kochi. 2009. pp. 85-88

[16] Dence JB, Dence TP. Elements of the Theory of Numbers. USA: Academic

*DOI: http://dx.doi.org/10.5772/intechopen.93356*

[17] Apostol TM. Introduction to Analytic Number Theory. USA:

*A New Integer-to-Integer Transform*

10.1109/TIP.2014.2362652

[18] Lakshmi PG, Dominic S. Walsh– Hadamard transform kernel-based feature vector for shot boundary detection. IEEE Transactions on Image Processing. 2014;**23**(12):5187-5197. DOI:

[19] Tang M, Chen X, Wen J, Han Y. Hadamard transform-based optimized HEVC video coding. IEEE Transactions on Circuits and Systems for Video Technology. 2019;**29**(3):827-839. DOI:

features in particle filter framework for underwater object tracking. IEEE Transactions on Industrial Informatics. 2020;**16**(9):5712-5722. DOI: 10.1109/

10.1109/TCSVT.2018.2810324

[20] Rout DK, Subudhi BN, Veerakumar T, Chaudhury S. Walsh–Hadamard-kernel-based

TII.2019.2937902

**25**

Press; 1999

Springer; 1976

[9] Fino J. Relations between Haar and Walsh/Hadamard transforms. Proceedings of the IEEE. 1972;**60**: 647-648. DOI: 10.1109/PROC.1972.8719

[10] Roy RC. Relationship between the Haar transform and the MRT. In: Proceedings of the Eighth International Conference on Information Communication and Signal Processing (ICICS '11), December 2011. Singapore: IEEE; 2011. DOI: 10.1109/ ICICS.2011.6173127

[11] Roy RC. The relationship between the Hadamard transform and the MRT. In: Proceedings of the National Conference on Power Instrumentation Control and Computing (PICC '10), Thrissur. 2010

[12] Pei S, Chang K. Integer 2-D discrete Fourier transform pairs and eigenvectors using Ramanujan's sum. IEEE Signal Processing Letters. 2016;**23– 1**:70-74. DOI: 10.1109/LSP.2015.2501421

[13] Qiu Q, Thompson A, Calderbank R, Sapiro G. Data representation using the Weyl transform. IEEE Transactions on Signal Processing. 2016;**64**(7): 1844-1853. DOI: 10.1109/ TSP.2015.2505661

[14] Kamisli F. Lossless image and intraframe compression with integer-tointeger DST. IEEE Transactions on Circuits and Systems for Video Technology. 2019;**29**(2):502-516. DOI: 10.1109/TCSVT.2017.2787638

[15] Roy RC. Development of a new transform: MRT [thesis]. Cochin University of Science and Technology; 2010

*A New Integer-to-Integer Transform DOI: http://dx.doi.org/10.5772/intechopen.93356*

[16] Dence JB, Dence TP. Elements of the Theory of Numbers. USA: Academic Press; 1999

[17] Apostol TM. Introduction to Analytic Number Theory. USA: Springer; 1976

**References**

[1] Ahmed N et al. Discrete cosine transform. IEEE Transactions on Computers. 1974;**C-23**:90-93. DOI:

*Number Theory and Its Applications*

Proceedings of the all India Seminar on Innovative Electronics Technology and Futuristic Communication, Kochi. 2009.

[9] Fino J. Relations between Haar and

[10] Roy RC. Relationship between the Haar transform and the MRT. In: Proceedings of the Eighth International

Communication and Signal Processing (ICICS '11), December 2011. Singapore:

[11] Roy RC. The relationship between the Hadamard transform and the MRT.

Conference on Power Instrumentation Control and Computing (PICC '10),

[12] Pei S, Chang K. Integer 2-D discrete

[13] Qiu Q, Thompson A, Calderbank R, Sapiro G. Data representation using the Weyl transform. IEEE Transactions on

[14] Kamisli F. Lossless image and intraframe compression with integer-tointeger DST. IEEE Transactions on Circuits and Systems for Video Technology. 2019;**29**(2):502-516. DOI:

eigenvectors using Ramanujan's sum. IEEE Signal Processing Letters. 2016;**23– 1**:70-74. DOI: 10.1109/LSP.2015.2501421

In: Proceedings of the National

Fourier transform pairs and

Signal Processing. 2016;**64**(7): 1844-1853. DOI: 10.1109/ TSP.2015.2505661

10.1109/TCSVT.2017.2787638

2010

[15] Roy RC. Development of a new transform: MRT [thesis]. Cochin University of Science and Technology;

Thrissur. 2010

Walsh/Hadamard transforms. Proceedings of the IEEE. 1972;**60**: 647-648. DOI: 10.1109/PROC.1972.8719

Conference on Information

IEEE; 2011. DOI: 10.1109/ ICICS.2011.6173127

pp. 85-88

[2] Bracewell RN. Discrete Hartley transform. Journal of the Optical Society of America. 1983;**73**:1832-1835. DOI:

[4] Roy RC, Gopikakumari R. A new transform for 2-d signal representation (MRT) and some of its properties. In: Proceedings of the International Conference in Signal Processing and Communication (SPCOM '04). Bangalore: IEEE; 2004. pp. 363-367. DOI: 10.1109/SPCOM.2004.1458423

[5] Lun DPK, Hsung TC, Shen TW. Orthogonal discrete periodic Radon transform. Part I: Theory and

[6] Roy RC, Anish Kumar MS, Gopikakumari R. An invertible

Proceedings of the International

[7] Anish Kumar MS, Roy RC, Gopikakumari R. A new transform coder for gray scale images using 44 MRT. Elsevier AEU—International

Journal of Electronics and

ISSPA.2007.4555504

00498-X

**24**

realization. Signal Processing. 2003;**83**: 941-955. DOI: 10.1016/S0165-1684(02)

transform for image representation and its application to image compression. In:

Symposium in Signal Processing (ISSPA '07). Sharjah: IEEE; 2007. DOI: 10.1109/

Communications. 2008;**62–8**:627-630. DOI: 10.1016/j.aeue.2007.04.009

[8] Roy RC, Gopikakumari R. 1-D MRT: A new transform for 1-D signals. In:

10.1109/T-C.1974.223784

10.1364/JOSA.73.001832

[3] Ersoy OK. A two-stage representation of DFT and its applications. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1987;**35**:825-831. DOI: 10.1109/TASSP.1987.1165202

[18] Lakshmi PG, Dominic S. Walsh– Hadamard transform kernel-based feature vector for shot boundary detection. IEEE Transactions on Image Processing. 2014;**23**(12):5187-5197. DOI: 10.1109/TIP.2014.2362652

[19] Tang M, Chen X, Wen J, Han Y. Hadamard transform-based optimized HEVC video coding. IEEE Transactions on Circuits and Systems for Video Technology. 2019;**29**(3):827-839. DOI: 10.1109/TCSVT.2018.2810324

[20] Rout DK, Subudhi BN, Veerakumar T, Chaudhury S. Walsh–Hadamard-kernel-based features in particle filter framework for underwater object tracking. IEEE Transactions on Industrial Informatics. 2020;**16**(9):5712-5722. DOI: 10.1109/ TII.2019.2937902

**Chapter 2**

**Abstract**

*Samin Riasat*

**1. Introduction**

Robbins identity [1, 2]:

Woods-Robbins identity.

**27**

of the form

Digit Sums and Infinite Products

Consider the sequence *un* defined as follows: *un* ¼ þ1 if the sum of the base *b* digits of *n* is even, and *un* ¼ �1 otherwise, where we take *b* ¼ 2. Recall that the Woods-Robbins infinite product involves a rational function in *n* and the sequence *un*. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in *n* and the sequence *un* was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of *b*. We illustrate the approach by evaluating a large class of similar infinite products.

**Keywords:** radix representations, digit sums, Prouhet-Thue-Morse sequence,

Throughout this chapter *n* will denote a non-negative integer. Let *sb*ð Þ *n* denote the sum of the base *<sup>b</sup>* digits of *<sup>n</sup>*, and put *un* ¼ �ð Þ<sup>1</sup> *<sup>s</sup>*2ð Þ *<sup>n</sup>* . We study infinite products

> *n* þ *b n* þ *c* � �*un*

> > 2

¼ 1 ffiffi 2

¼ *f* 0,

1 2

*:* (1)

<sup>p</sup> , which is the famous Woods-

p *:* (2)

� �*:* (3)

Woods and Robbins product, closed formulas for infinite products

*f b*ð Þ ,*<sup>c</sup>* <sup>≔</sup>Y<sup>∞</sup>

Y∞ *n*¼0

only known nontrivial value of *<sup>f</sup>* is *<sup>f</sup>*ð Þ¼ <sup>1</sup>*=*2, 1 ffiffi

known (see, e.g., [5, 6]) about the similar product

Y∞ *n*¼1 *n*¼1

(We show in Section 2 that *f b*ð Þ ,*c* converges for *b*,*c*∈ n �f g 1, �2, �3, … ). Plainly *f b*ð Þ¼ ,*c* 1*=f c*ð Þ , *b* and *f b*ð Þ¼ , *b* 1. Up to these relations, it seems that the

> 2*n* þ 1 2*n* þ 2 � �*un*

2*n* 2*n* þ 1 � �*un*

The material in this chapter is based on the two papers [7, 8].

Several infinite products inspired by it were discovered afterwards (see, e.g., [3, 4]). But none of them involve the sequence *un*. Moreover, almost nothing is

Our goal is to study these infinite products in detail. This will allow us to gain a deeper understanding of such products as well as evaluate more products like the

## **Chapter 2** Digit Sums and Infinite Products

*Samin Riasat*

#### **Abstract**

Consider the sequence *un* defined as follows: *un* ¼ þ1 if the sum of the base *b* digits of *n* is even, and *un* ¼ �1 otherwise, where we take *b* ¼ 2. Recall that the Woods-Robbins infinite product involves a rational function in *n* and the sequence *un*. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in *n* and the sequence *un* was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of *b*. We illustrate the approach by evaluating a large class of similar infinite products.

**Keywords:** radix representations, digit sums, Prouhet-Thue-Morse sequence, Woods and Robbins product, closed formulas for infinite products

#### **1. Introduction**

Throughout this chapter *n* will denote a non-negative integer. Let *sb*ð Þ *n* denote the sum of the base *<sup>b</sup>* digits of *<sup>n</sup>*, and put *un* ¼ �ð Þ<sup>1</sup> *<sup>s</sup>*2ð Þ *<sup>n</sup>* . We study infinite products of the form

$$f(b,c) \coloneqq \prod\_{n=1}^{\infty} \left( \frac{n+b}{n+c} \right)^{u\_n}.\tag{1}$$

(We show in Section 2 that *f b*ð Þ ,*c* converges for *b*,*c*∈ n �f g 1, �2, �3, … ).

Plainly *f b*ð Þ¼ ,*c* 1*=f c*ð Þ , *b* and *f b*ð Þ¼ , *b* 1. Up to these relations, it seems that the only known nontrivial value of *<sup>f</sup>* is *<sup>f</sup>*ð Þ¼ <sup>1</sup>*=*2, 1 ffiffi 2 <sup>p</sup> , which is the famous Woods-Robbins identity [1, 2]:

$$\prod\_{n=0}^{\infty} \left(\frac{2n+1}{2n+2}\right)^{u\_n} = \frac{1}{\sqrt{2}}.\tag{2}$$

Several infinite products inspired by it were discovered afterwards (see, e.g., [3, 4]). But none of them involve the sequence *un*. Moreover, almost nothing is known (see, e.g., [5, 6]) about the similar product

$$\prod\_{n=1}^{\infty} \left(\frac{2n}{2n+1}\right)^{u\_n} = f\left(0, \frac{1}{2}\right). \tag{3}$$

Our goal is to study these infinite products in detail. This will allow us to gain a deeper understanding of such products as well as evaluate more products like the Woods-Robbins identity.

The material in this chapter is based on the two papers [7, 8].

#### **2. General properties of the function** *f*

First we establish a general result from [7] on convergence.

**Lemma 1.1** Let *R*∈ ð Þ *X* be a rational function such that the values *R n*ð Þ are defined and nonzero for *n*≥1. Then, the infinite product Q *<sup>n</sup>R n*ð Þ*un* converges if and only if the numerator and the denominator of *R* have the same degree and same leading coefficient.

**Proof.** If the infinite product converges, then *R n*ð Þ must tend to 1 when *n* tends to infinity. Thus the numerator and the denominator of *R* have the same degree and the same leading coefficient.

Now suppose that the numerator and the denominator of *R* have the same leading coefficient and the same degree. Decomposing them in factors of degree 1, it suffices, for proving that the infinite product converges, to show that infinite products of the form

$$\prod\_{n=1}^{\infty} \left( \frac{n+b}{n+c} \right)^{u\_n} \tag{4}$$

Hence we can rewrite property 3 as

*DOI: http://dx.doi.org/10.5772/intechopen.92365*

*Digit Sums and Infinite Products*

be completely evaluated in terms of the other.

about monotonic/continuous/smooth solutions?

**Theorem 1.1** The following relations hold.

Y∞ *n*¼1

> Y∞ *n*¼0

Y∞ *n*¼1

> Y∞ *n*¼1

1.For *b*,*c*∈ n �f g 1, �2, �3, … ,

2.For *b*,*c*∈ nf g 0, �1, �2, … ,

3.For *b*∈ n �f g 1, �2, �3, … ,

4.For *c*∈ n �f g 1, �2, �3, … ,

**Proof.**

**29**

2. As above.

Taking *<sup>c</sup>* <sup>¼</sup> *<sup>b</sup>* <sup>þ</sup> <sup>1</sup>

**3. Infinite products**

**3.1 Direct approach**

*f b*ð Þ¼ ,*<sup>c</sup> <sup>f</sup> <sup>b</sup>*

Thus *f b*ð Þ ,*<sup>c</sup>* can be computed using only the quantities *h x*ð Þ¼ *<sup>f</sup> <sup>x</sup>*

*f b*ð Þ¼ ,*c*

*h b*ð Þ¼ *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> *<sup>b</sup>* <sup>þ</sup> <sup>3</sup> 2

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*þ<sup>1</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup> <sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*þ<sup>1</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*þ<sup>1</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup> <sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*þ<sup>1</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*þ<sup>1</sup>

� � !*un*

*<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*þ<sup>3</sup> 4 � � *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*

> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*

1. This follows immediately using properties 1–3 in Section 2.

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup> <sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*þ<sup>1</sup>

� � !*un*

2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*

� � !*un*

2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*

> 2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*

� � !*un*

2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*

4 � �

2

2 � �

2

2 � �

<sup>¼</sup> *<sup>c</sup>* <sup>þ</sup> <sup>1</sup> *b* þ 1

*:* (12)

¼ 1*:* (13)

<sup>2</sup>ð Þ *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> *:* (14)

¼ *c* þ 1*:* (15)

2

2 � �

2

<sup>¼</sup> *<sup>b</sup>* <sup>þ</sup> <sup>3</sup>

<sup>2</sup> , *<sup>b</sup>*þ<sup>1</sup> 2 � � *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> *<sup>=</sup> <sup>f</sup> <sup>c</sup>*

> *c* þ 1 *b* þ 1 � *h b*ð Þ

So understanding *f* is equivalent to understanding *h*, in the sense that each can

*h b* þ

Similar questions can be asked for *h*: is it the unique solution to Eq. (11)? What

1 2

<sup>2</sup> in Eq. (10) gives the functional equation:

<sup>2</sup> , *<sup>c</sup>*þ<sup>1</sup> 2 � �

*<sup>c</sup>* <sup>þ</sup> <sup>1</sup> *:* (9)

*h c*ð Þ *:* (10)

� �*h*ð Þ <sup>2</sup>*<sup>b</sup> :* (11)

<sup>2</sup> , *<sup>x</sup>*þ<sup>1</sup> 2 � �, via

converge for complex numbers *b* and *c* such that *n* þ *b* and *n* þ *c* do not vanish for any *n*≥ 1. Since the general factor of such a product tends to 1, it is equivalent, grouping the factors pairwise, to proving that the product

$$\prod\_{n=1}^{\infty} \left[ \left( \frac{2n+b}{2n+c} \right)^{u\_{2n}} \left( \frac{2n+1+b}{2n+1+c} \right)^{u\_{2n+1}} \right] \tag{5}$$

converges. Since *u*2*<sup>n</sup>* ¼ *un* and *u*2*n*þ<sup>1</sup> ¼ �*un*, we only need to prove that the infinite product

$$\prod\_{n=1}^{\infty} \left( \frac{(2n+b)(2n+1+c)}{(2n+c)(2n+1+b)} \right)^{u\_n} \tag{6}$$

converges. Taking (the principal determination of) logarithms, we see that

$$\log\left(\frac{(2n+b)(2n+1+c)}{(2n+c)(2n+1+b)}\right) = O\left(\frac{1}{n^2}\right) \tag{7}$$

which gives the convergence result.

Hence *f b*ð Þ ,*c* converges for any *b*,*c*∈ n �f g 1, �2, �3, … . Using the definition of *un*, it follows that for any *b*,*c*, *d*∈ n �f g 1, �2, �3, … ,

1. *f b*ð Þ¼ , *b* 1.

$$2. f(b, c)f(c, d) = f(b, d).$$

$$3. f(b,c) = \frac{c+1}{b+1} f\left(\frac{b}{2}, \frac{c}{2}\right) f\left(\frac{c+1}{2}, \frac{b+1}{2}\right).$$

One can ask the natural question: is *f* the unique function satisfying these properties?

#### **2.1 A new function**

Properties 1 and 2 above give

$$f(b,c)f(d,e) = \frac{f(b,c)f(c,d)f(d,e)f(d,c)}{f(c,d)f(d,c)} = \frac{f(b,e)f(d,c)}{f(c,c)} = f(b,e)f(d,c). \tag{8}$$

Hence we can rewrite property 3 as

**2. General properties of the function** *f*

*Number Theory and Its Applications*

leading coefficient.

products of the form

infinite product

1. *f b*ð Þ¼ , *b* 1.

3. *f b*ð Þ¼ ,*<sup>c</sup> <sup>c</sup>*þ<sup>1</sup>

**2.1 A new function**

**28**

2. *f b*ð Þ ,*c f c*ð Þ¼ , *d f b*ð Þ , *d* .

*<sup>b</sup>*þ<sup>1</sup> *<sup>f</sup> <sup>b</sup>* <sup>2</sup> , *<sup>c</sup>* 2 � �*f <sup>c</sup>*þ<sup>1</sup>

Properties 1 and 2 above give

*f b*ð Þ ,*<sup>c</sup> f d*ð Þ¼ ,*<sup>e</sup> f b*ð Þ ,*<sup>c</sup> f c*ð Þ , *<sup>d</sup> f d*ð Þ ,*<sup>e</sup> f d*ð Þ ,*<sup>c</sup>*

the same leading coefficient.

First we establish a general result from [7] on convergence.

defined and nonzero for *n*≥1. Then, the infinite product Q

grouping the factors pairwise, to proving that the product

Y∞ *n*¼1

of *un*, it follows that for any *b*,*c*, *d*∈ n �f g 1, �2, �3, … ,

<sup>2</sup> , *<sup>b</sup>*þ<sup>1</sup> 2 � �.

which gives the convergence result.

2*n* þ *b* 2*n* þ *c*

Y∞ *n*¼1

**Lemma 1.1** Let *R*∈ ð Þ *X* be a rational function such that the values *R n*ð Þ are

**Proof.** If the infinite product converges, then *R n*ð Þ must tend to 1 when *n* tends to infinity. Thus the numerator and the denominator of *R* have the same degree and

only if the numerator and the denominator of *R* have the same degree and same

Now suppose that the numerator and the denominator of *R* have the same leading coefficient and the same degree. Decomposing them in factors of degree 1, it suffices, for proving that the infinite product converges, to show that infinite

> *n* þ *b n* þ *c* � �*un*

converge for complex numbers *b* and *c* such that *n* þ *b* and *n* þ *c* do not vanish for any *n*≥ 1. Since the general factor of such a product tends to 1, it is equivalent,

� �*<sup>u</sup>*2*n*þ<sup>1</sup> � �

2*n* þ 1 þ *c*

<sup>¼</sup> *<sup>O</sup>* <sup>1</sup> *n*2 � �

*f c*ð Þ ,*<sup>c</sup>* <sup>¼</sup> *f b*ð Þ ,*<sup>e</sup> f d*ð Þ ,*<sup>c</sup> :* (8)

� �*<sup>u</sup>*2*<sup>n</sup>* <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>b</sup>*

converges. Since *u*2*<sup>n</sup>* ¼ *un* and *u*2*n*þ<sup>1</sup> ¼ �*un*, we only need to prove that the

converges. Taking (the principal determination of) logarithms, we see that

Hence *f b*ð Þ ,*c* converges for any *b*,*c*∈ n �f g 1, �2, �3, … . Using the definition

One can ask the natural question: is *f* the unique function satisfying these properties?

*f c*ð Þ , *<sup>d</sup> f d*ð Þ ,*<sup>c</sup>* <sup>¼</sup> *f b*ð Þ ,*<sup>e</sup> f d*ð Þ ,*<sup>c</sup>*

log ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>c</sup>* ð Þ 2*n* þ *c* ð Þ 2*n* þ 1 þ *b* � �

ð Þ 2*n* þ *b* ð Þ 2*n* þ 1 þ *c* ð Þ 2*n* þ *c* ð Þ 2*n* þ 1 þ *b* � �*un*

Y∞ *n*¼1 *<sup>n</sup>R n*ð Þ*un* converges if and

(4)

(5)

(6)

(7)

$$f(b,c) = \frac{f\left(\frac{b}{2}, \frac{b+1}{2}\right)}{b+1} / \frac{f\left(\frac{c}{2}, \frac{c+1}{2}\right)}{c+1} . \tag{9}$$

Thus *f b*ð Þ ,*<sup>c</sup>* can be computed using only the quantities *h x*ð Þ¼ *<sup>f</sup> <sup>x</sup>* <sup>2</sup> , *<sup>x</sup>*þ<sup>1</sup> 2 � �, via

$$f(b,c) = \frac{c+1}{b+1} \cdot \frac{h(b)}{h(c)}.\tag{10}$$

So understanding *f* is equivalent to understanding *h*, in the sense that each can be completely evaluated in terms of the other.

Taking *<sup>c</sup>* <sup>¼</sup> *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> in Eq. (10) gives the functional equation:

$$h(b) = \frac{b+1}{b+\frac{3}{2}}h\left(b+\frac{1}{2}\right)h(2b). \tag{11}$$

Similar questions can be asked for *h*: is it the unique solution to Eq. (11)? What about monotonic/continuous/smooth solutions?

#### **3. Infinite products**

#### **3.1 Direct approach**

**Theorem 1.1** The following relations hold.

1.For *b*,*c*∈ n �f g 1, �2, �3, … , Y∞ *n*¼1 ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*þ<sup>1</sup> 2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup>* 2 � � ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup> <sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*þ<sup>1</sup> 2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>* 2 � � !*un* <sup>¼</sup> *<sup>c</sup>* <sup>þ</sup> <sup>1</sup> *b* þ 1 *:* (12)

$$\begin{aligned} \text{2. For } b, c \in \mathbb{C} \backslash \{0, -1, -2, \dots\},\\ \prod\_{n=0}^{\infty} \left( \frac{(n+b)\left(n + \frac{b+1}{2}\right)\left(n + \frac{c}{2}\right)}{(n+c)\left(n + \frac{c+1}{2}\right)\left(n + \frac{b}{2}\right)} \right)^{u\_n} = \mathbf{1}. \end{aligned} \tag{13}$$

3.For *b*∈ n �f g 1, �2, �3, … ,

$$\prod\_{n=1}^{\infty} \left( \frac{(n+b)\left(n+\frac{b+1}{4}\right)}{\left(n+\frac{b+3}{4}\right)\left(n+\frac{b}{2}\right)} \right)^{u\_n} = \frac{b+3}{2(b+1)}.\tag{14}$$

4.For *c*∈ n �f g 1, �2, �3, … , Y∞ *n*¼1 *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � � *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup>* 2 � � ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>c</sup> <sup>n</sup>* <sup>þ</sup> *<sup>c</sup>*þ<sup>1</sup> 2 � � !*un* ¼ *c* þ 1*:* (15)

#### **Proof.**

1. This follows immediately using properties 1–3 in Section 2.

2. As above.


**Corollary 1.1** For any positive rational number *q*, there exist monic polynomials *P*, *Q* ∈ <sup>1</sup> <sup>4</sup> ℤ½ � *X* , both at most cubic, such that

$$\prod\_{n=1}^{\infty} \left( \frac{P(n)}{Q(n)} \right)^{u\_n} = q. \tag{16}$$

Taking *b* ¼ 1 in Eq. (11) gives

*DOI: http://dx.doi.org/10.5772/intechopen.92365*

hence *<sup>h</sup>*ð Þ <sup>3</sup>*=*<sup>2</sup> *<sup>h</sup>*ð Þ¼ <sup>2</sup> <sup>5</sup> ffiffi

*Digit Sums and Infinite Products*

which is equivalent to

**4. Some analytical results**

**Lemma 1.2** Let *b*,*c*∈ð Þ �1, ∞ .

1. If *b* ¼ *c*, then *f b*ð Þ¼ ,*c* 1.

2. If *b*>*c*, then

3. If *b*< *c*, then

Let *b*>*c*> � 1, and put

*an* <sup>¼</sup> log *<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*

*n* þ *c*

and *h*.

parts,

**31**

*h*ð Þ¼ 1

<sup>p</sup> *<sup>=</sup>*4, and this gives

Taking *b* ¼ 3*=*2 in Eq. (11) and using the previous result gives

*<sup>h</sup>*ð Þ<sup>2</sup> <sup>2</sup>

ð Þ 2*n* þ 2 ð Þ *n* þ 1 ð Þ 2*n* þ 3 ð Þ *n* þ 2 � �*un*

These identities can be also combined in pairs to obtain other identities.

*c* þ 1 *b* þ 1 � �<sup>2</sup>

1<*f b*ð Þ ,*c* <

**Proof.** Using properties 1–3 from Section 2, it suffices to prove 2.

� �, *SN* <sup>¼</sup> <sup>X</sup>

*c* þ 1 *b* þ 1 � �<sup>2</sup>

*N*

*anun*, *UN* <sup>¼</sup> <sup>X</sup>

*n*¼1

Note that *an* is positive and strictly decreasing to 0. Using *s*2ð Þ¼ 2*n* þ 1 1 þ *s*2ð Þ 2*n* , it follows that *Un* ∈f g �2, �1, 0 and *Un* � *n*ð Þ mod2 , for each *n*. Using summation by

We saw in the previous section that some of the infinite products we evaluated were integers, some were rational, and some were quadratic irrational. In the hope of further understanding their nature, we now study the analytical behaviors of *f*

*<sup>h</sup>*ð Þ¼ <sup>3</sup> <sup>3</sup>

ffiffi 2

2

Y∞ *n*¼0

> Y∞ *n*¼0

4 5 *<sup>h</sup>* <sup>3</sup> 2

ð Þ 4*n* þ 3 ð Þ 2*n* þ 2 ð Þ 4*n* þ 5 ð Þ 2*n* þ 3 � �*un*

� �*h*ð Þ<sup>2</sup> ; (24)

p *:* (25)

p *:* (27)

p (26)

<*f b*ð Þ ,*c* <1*:* (28)

*:* (29)

*un:* (30)

*N*

*n*¼1

¼ 1 ffiffi 2

¼ 1 ffiffi 2

Furthermore, if *q* is an integer, then *P* and *Q* can be chosen to be at most quadratic.

We still do not know exactly which numbers are given by such infinite products.

#### **3.2 Functional equation approach**

Recall the functional Eq. (11):

$$h(b) = \frac{b+1}{b+\frac{3}{2}}h\left(b+\frac{1}{2}\right)h(2b). \tag{17}$$

Taking *b* ¼ 0 in Eq. (11) gives

$$h(\mathbf{0}) = \frac{2}{3}h\left(\frac{1}{2}\right)h(\mathbf{0}),\tag{18}$$

i.e., *h*ð Þ¼ 1*=*2 3*=*2. This shows that

$$\prod\_{n=0}^{\infty} \left(\frac{4n+3}{4n+1}\right)^{u\_n} = 2.\tag{19}$$

Next, taking *b* ¼ 1*=*2 in Eq. (11) gives

$$h\left(\frac{1}{2}\right) = \frac{3}{4}h(\mathbf{1})^2;\tag{20}$$

hence *<sup>h</sup>*ð Þ¼ <sup>1</sup> ffiffi 2 <sup>p</sup> , and we recover the Woods-Robbins identity

$$\prod\_{n=0}^{\infty} \left(\frac{2n+2}{2n+1}\right)^{u\_n} = \sqrt{2}.\tag{21}$$

Similarly, taking *b* ¼ �1*=*2 in Eq. (11) gives

$$h\left(-\frac{1}{2}\right) = \frac{1}{2}h(0)h(-1) = \frac{1}{2}f\left(0, \frac{1}{2}\right)f\left(-\frac{1}{2}, 0\right) = \frac{1}{2}f\left(-\frac{1}{2}, \frac{1}{2}\right),\tag{22}$$

i.e.,

$$\prod\_{n=1}^{\infty} \left( \frac{(4n-1)(2n+1)}{(4n+1)(2n-1)} \right)^{u\_n} = \frac{1}{2}.\tag{23}$$

Taking *b* ¼ 1 in Eq. (11) gives

3. Take *c* ¼ ð Þ *b* þ 1 *=*2 in Eq. (12).

<sup>4</sup> ℤ½ � *X* , both at most cubic, such that

Y∞ *n*¼1

*h b*ð Þ¼ *<sup>b</sup>* <sup>þ</sup> <sup>1</sup> *<sup>b</sup>* <sup>þ</sup> <sup>3</sup> 2

*h*ð Þ¼ 0

Y∞ *n*¼0

> *<sup>h</sup>* <sup>1</sup> 2 � �

Y∞ *n*¼0

> 1 2

2 3 *<sup>h</sup>* <sup>1</sup> 2 � �

4*n* þ 3 4*n* þ 1 � �*un*

> ¼ 3 4 *<sup>h</sup>*ð Þ<sup>1</sup> <sup>2</sup>

<sup>p</sup> , and we recover the Woods-Robbins identity

<sup>¼</sup> ffiffi 2

> ¼ 1 2 *<sup>f</sup>* � <sup>1</sup> 2 , 1 2 � �

¼ 1 2

*<sup>f</sup>* � <sup>1</sup> 2 , 0 � �

2*n* þ 2 2*n* þ 1 � �*un*

*<sup>f</sup>* 0, <sup>1</sup> 2 � �

ð Þ 4*n* � 1 ð Þ 2*n* þ 1 ð Þ 4*n* þ 1 ð Þ 2*n* � 1 � �*un*

**Corollary 1.1** For any positive rational number *q*, there exist monic polynomials

We still do not know exactly which numbers are given by such infinite products.

*h b* þ

1 2 � �

¼ *q:* (16)

*h*ð Þ 2*b :* (17)

*h*ð Þ 0 , (18)

¼ 2*:* (19)

; (20)

<sup>p</sup> *:* (21)

, (22)

*:* (23)

*P n*ð Þ *Q n*ð Þ � �*un*

Furthermore, if *q* is an integer, then *P* and *Q* can be chosen to be at most

4. Take *b* ¼ 0 in Eq. (12).

*Number Theory and Its Applications*

**3.2 Functional equation approach**

Recall the functional Eq. (11):

Taking *b* ¼ 0 in Eq. (11) gives

i.e., *h*ð Þ¼ 1*=*2 3*=*2. This shows that

Next, taking *b* ¼ 1*=*2 in Eq. (11) gives

2

¼ 1 2

Similarly, taking *b* ¼ �1*=*2 in Eq. (11) gives

*h*ð Þ 0 *h*ð Þ¼ �1

Y∞ *n*¼1

hence *<sup>h</sup>*ð Þ¼ <sup>1</sup> ffiffi

*<sup>h</sup>* � <sup>1</sup> 2 � �

i.e.,

**30**

*P*, *Q* ∈ <sup>1</sup>

quadratic.

$$h(\mathbf{1}) = \frac{4}{5}h\left(\frac{3}{2}\right)h(\mathbf{2});\tag{24}$$

hence *<sup>h</sup>*ð Þ <sup>3</sup>*=*<sup>2</sup> *<sup>h</sup>*ð Þ¼ <sup>2</sup> <sup>5</sup> ffiffi 2 <sup>p</sup> *<sup>=</sup>*4, and this gives

$$\prod\_{n=0}^{\infty} \left( \frac{(4n+3)(2n+2)}{(4n+5)(2n+3)} \right)^{u\_n} = \frac{1}{\sqrt{2}}.\tag{25}$$

Taking *b* ¼ 3*=*2 in Eq. (11) and using the previous result gives

$$h(\mathbf{2})^2 h(\mathbf{3}) = \frac{\mathbf{3}}{\sqrt{2}}\tag{26}$$

which is equivalent to

$$\prod\_{n=0}^{\infty} \left( \frac{(2n+2)(n+1)}{(2n+3)(n+2)} \right)^{u\_n} = \frac{1}{\sqrt{2}}.\tag{27}$$

These identities can be also combined in pairs to obtain other identities.

#### **4. Some analytical results**

We saw in the previous section that some of the infinite products we evaluated were integers, some were rational, and some were quadratic irrational. In the hope of further understanding their nature, we now study the analytical behaviors of *f* and *h*.

**Lemma 1.2** Let *b*,*c*∈ð Þ �1, ∞ .

1. If *b* ¼ *c*, then *f b*ð Þ¼ ,*c* 1.

2. If *b*>*c*, then

$$\left(\frac{c+1}{b+1}\right)^2 < f(b,c) < 1. \tag{28}$$

3. If *b*< *c*, then

$$1 < f(b, c) < \left(\frac{c + 1}{b + 1}\right)^2. \tag{29}$$

**Proof.** Using properties 1–3 from Section 2, it suffices to prove 2. Let *b*>*c*> � 1, and put

$$a\_n = \log\left(\frac{n+b}{n+c}\right), \quad \mathbb{S}\_N = \sum\_{n=1}^N a\_n u\_n, \quad U\_N = \sum\_{n=1}^N u\_n. \tag{30}$$

Note that *an* is positive and strictly decreasing to 0. Using *s*2ð Þ¼ 2*n* þ 1 1 þ *s*2ð Þ 2*n* , it follows that *Un* ∈f g �2, �1, 0 and *Un* � *n*ð Þ mod2 , for each *n*. Using summation by parts,

$$\mathbf{S}\_{N} = a\_{N+1}\mathbf{U}\_{N} + \sum\_{n=1}^{N} \mathbf{U}\_{n}(a\_{n} - a\_{n+1}).\tag{31}$$

as *M* ! ∞, for any *x*∈ð Þ �2, ∞ and *N* > *M*. Thus *S*<sup>0</sup>

*k*! X *N*

*n*¼1 *un*

X *N*

*n*¼*M*þ1

X *N*

*n*¼*M*þ1

*k*! ð Þ <sup>2</sup>*<sup>M</sup>* � <sup>1</sup> *<sup>k</sup>*þ<sup>1</sup> ! <sup>0</sup>

Therefore, by induction, *h* has derivatives of all orders on ð Þ �2, ∞ .

ð Þ �<sup>1</sup> *<sup>k</sup>*�<sup>1</sup> *k*

*k*¼1

*<sup>H</sup>*ð Þ *<sup>k</sup>*þ<sup>1</sup> ð Þ¼ � *<sup>x</sup>* ð Þ<sup>1</sup> *<sup>k</sup>*

X∞ *n*¼2

polynomial for *H x*ð Þ of degree *k* is absolutely bounded above by

X∞ *n*¼2

<sup>¼</sup> *<sup>k</sup>*! ð Þ 2*M* þ 1 þ *x*

*S* ð Þ *k*þ1 *<sup>N</sup>* ¼ �ð Þ<sup>1</sup> *<sup>k</sup>*

*Digit Sums and Infinite Products*

*DOI: http://dx.doi.org/10.5772/intechopen.92365*

ð Þ *k*þ1 *M*

� � �≤*k*!

≤*k*!

<

ð Þ �2, <sup>∞</sup> , i.e., *<sup>h</sup>*ð Þ*<sup>k</sup>* is differentiable on ð Þ �2, <sup>∞</sup> .

log *h x*ð Þ¼ log *h a*ð ÞþX<sup>∞</sup>

**Proof.** Let *H x*ð Þ¼ log *h x*ð Þ. By Theorem 1.5,

<sup>∣</sup>*H*ð Þ *<sup>k</sup>*þ<sup>1</sup> ð Þ *<sup>x</sup>* ∣ ≤*k*!

1 *k* þ 1

Taylor expansion about *a* for *x* in the given range.

**5. Further remarks on** *h*ð Þ **0**

**Theorem 1.6** Let *a*≥0. Then

for *x*∈ ½ � *a* � 1, *a* þ 1 .

Hence

**33**

As before,

*S* ð Þ *k*þ1 *<sup>N</sup>* � *S*

� � �

ð Þ �2, ∞ , which shows that log *h*, hence *h*, is differentiable on ð Þ �2, ∞ .

Now suppose that derivatives of *h* up to order *k* exist for some *k*≥ 1. Note that

1 ð Þ 2*n* þ *x*

1 ð Þ 2*n* � 1 þ *x*

as *<sup>M</sup>* ! <sup>∞</sup>, for any *<sup>x</sup>*∈ð Þ �2, <sup>∞</sup> and *<sup>N</sup>* <sup>&</sup>gt; *<sup>M</sup>*. Hence *<sup>S</sup>*ð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>n</sup>* converges uniformly on

*k*! X∞ *n*¼2

*<sup>k</sup>*þ<sup>1</sup> <sup>≤</sup> *<sup>k</sup>*!

1 j j *n* þ *x*

for *x*∈ ½ � *a* � 1, *a* þ 1 . So by Taylor's inequality, the remainder for the Taylor

1 ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>a</sup>* � <sup>1</sup> *<sup>k</sup>*þ<sup>1</sup>

which tends to 0 as *k* ! ∞, since *a*≥0 and ∣*x* � *a*∣ ≤1. Therefore *H x*ð Þ equals its

As mentioned in Section 1, not much is known about the quantity *h*ð Þ 0 ≈1*:*62816. We give the following explanation as to why *h*ð Þ 0 might behave specially in a sense.

!

X∞ *n*¼2

*<sup>k</sup>*þ<sup>1</sup> � *<sup>k</sup>*!

1 ð Þ 2*n* þ *x*

*<sup>k</sup>*þ<sup>1</sup> � <sup>1</sup>

!

*<sup>k</sup>*þ<sup>1</sup> � <sup>1</sup>

!

ð Þ 2*N* þ 1 þ *x*

*un* ð Þ *n* þ *a k*

*un* ð Þ *n* þ *x*

> X∞ *n*¼2

1

!

!

ð Þ 2*n* þ 1 þ *x*

ð Þ 2*n* þ 1 þ *x*

*<sup>k</sup>*þ<sup>1</sup> � <sup>1</sup>

*<sup>n</sup>* converges uniformly on

*k*þ1

*k*þ1

*k*þ1

ð Þ *<sup>x</sup>* � *<sup>a</sup> <sup>k</sup>* (40)

*<sup>k</sup>*þ<sup>1</sup> *:* (41)

ð Þ *<sup>n</sup>* <sup>þ</sup> *<sup>a</sup>* � <sup>1</sup> *<sup>k</sup>*þ<sup>1</sup> (42)

j j *<sup>x</sup>* � *<sup>a</sup> <sup>k</sup>*þ<sup>1</sup> (43)

ð Þ 2*n* þ 1 þ *x*

*k*þ1

*:* (38)

(39)

So �2*a*<sup>1</sup> <*SN* <0 for *N* ≥2. Exponentiating and taking *N* ! ∞ gives the desired result.

Lemma 1.2 together with Eq. (10) implies the following results.

**Theorem 1.2** *h x*ð Þ*=*ð Þ *x* þ 1 is strictly decreasing on ð Þ �1, ∞ , and *h x*ð Þð Þ *x* þ 1 is strictly increasing on ð Þ �1, ∞ .

**Proof.** Let �1<*b*<*c*. By Eqs. (29) and (10),

$$1 < \frac{c+1}{b+1} \cdot \frac{h(b)}{h(c)} < \left(\frac{c+1}{b+1}\right)^2\tag{32}$$

from which the result follows.

**Theorem 1.3** For *b*,*c*∈ð Þ �1, ∞ , *f b*ð Þ ,*c* is strictly decreasing in *b* and strictly increasing in *c*.

**Proof.** By Eq. (10),

$$\frac{h(c)f(b,c)}{c+1} = \frac{h(b)}{b+1} \tag{33}$$

and

$$\frac{(b+1)f(b,c)}{h(b)} = \frac{c+1}{h(c)}\tag{34}$$

hence the result follows from Theorem 1.2. **Theorem 1.4** For *x*∈ð Þ �2, ∞ ,

$$1 < h(\boldsymbol{x}) < \left(\frac{\boldsymbol{x} + \mathfrak{z}}{\boldsymbol{x} + \mathfrak{z}}\right)^2. \tag{35}$$

**Proof.** This follows from taking *b* ¼ *x=*2 and *c* ¼ ð Þ *x* þ 1 *=*2 in Eq. (10), then using Eq. (29).

We now prove some results on differentiability.

**Theorem 1.5** *h x*ð Þ is smooth on ð Þ �2, ∞ .

**Proof.** Take *b* ¼ *x=*2 and *c* ¼ ð Þ *x* þ 1 *=*2 in Eq. (30). Then the sequence *Sn* of smooth functions on ð Þ �2, ∞ converges pointwise to log *h*.

Differentiating with respect to *x* gives

$$S'\_N = \sum\_{n=1}^N \frac{u\_n}{(2n+\varkappa)(2n+1+\varkappa)} = \sum\_{n=1}^N u\_n \left(\frac{1}{2n+\varkappa} - \frac{1}{2n+1+\varkappa}\right). \tag{36}$$

Hence

$$\begin{aligned} \left| S\_N' - S\_M' \right| &\le \sum\_{n=M+1}^N \left( \frac{1}{2n+\varkappa} - \frac{1}{2n+1+\varkappa} \right) \\ &\le \sum\_{n=M+1}^N \left( \frac{1}{2n-1+\varkappa} - \frac{1}{2n+1+\varkappa} \right) \\ &= \frac{1}{2M+1+\varkappa} - \frac{1}{2N+1+\varkappa} \\ &< \frac{1}{2M-1} \to 0 \end{aligned} \tag{37}$$

#### *Digit Sums and Infinite Products DOI: http://dx.doi.org/10.5772/intechopen.92365*

as *M* ! ∞, for any *x*∈ð Þ �2, ∞ and *N* > *M*. Thus *S*<sup>0</sup> *<sup>n</sup>* converges uniformly on ð Þ �2, ∞ , which shows that log *h*, hence *h*, is differentiable on ð Þ �2, ∞ .

Now suppose that derivatives of *h* up to order *k* exist for some *k*≥ 1. Note that

$$\mathcal{S}\_N^{(k+1)} = (-1)^k k! \sum\_{n=1}^N u\_n \left( \frac{1}{(2n+\varkappa)^{k+1}} - \frac{1}{(2n+1+\varkappa)^{k+1}} \right). \tag{38}$$

As before,

*SN* <sup>¼</sup> *aN*þ<sup>1</sup>*UN* <sup>þ</sup><sup>X</sup>

Lemma 1.2 together with Eq. (10) implies the following results.

*c* þ 1 *b* þ 1 � *h b*ð Þ *h c*ð Þ <sup>&</sup>lt;

result.

increasing in *c*.

and

using Eq. (29).

*S*0 *<sup>N</sup>* <sup>¼</sup> <sup>X</sup> *N*

Hence

**32**

*n*¼1

**Proof.** By Eq. (10),

strictly increasing on ð Þ �1, ∞ .

*Number Theory and Its Applications*

from which the result follows.

**Proof.** Let �1<*b*<*c*. By Eqs. (29) and (10),

hence the result follows from Theorem 1.2.

We now prove some results on differentiability.

*un* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>x</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>¼</sup> <sup>X</sup>

smooth functions on ð Þ �2, ∞ converges pointwise to log *h*.

�<sup>≤</sup> <sup>X</sup> *N*

*n*¼*M*þ1

*n*¼*M*þ1

<sup>¼</sup> <sup>1</sup>

1 <sup>2</sup>*<sup>M</sup>* � <sup>1</sup> ! <sup>0</sup>

<

≤ X *N*

**Theorem 1.5** *h x*ð Þ is smooth on ð Þ �2, ∞ .

Differentiating with respect to *x* gives

*S*0 *<sup>N</sup>* � *S*<sup>0</sup> *M*

� � �

**Theorem 1.4** For *x*∈ð Þ �2, ∞ ,

1<

*N*

*n*¼1

So �2*a*<sup>1</sup> <*SN* <0 for *N* ≥2. Exponentiating and taking *N* ! ∞ gives the desired

**Theorem 1.2** *h x*ð Þ*=*ð Þ *x* þ 1 is strictly decreasing on ð Þ �1, ∞ , and *h x*ð Þð Þ *x* þ 1 is

**Theorem 1.3** For *b*,*c*∈ð Þ �1, ∞ , *f b*ð Þ ,*c* is strictly decreasing in *b* and strictly

*<sup>c</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> *h b*ð Þ

*h b*ð Þ <sup>¼</sup> *<sup>c</sup>* <sup>þ</sup> <sup>1</sup>

*x* þ 3 *x* þ 2 � �<sup>2</sup>

*h c*ð Þ*f b*ð Þ ,*c*

ð Þ *b* þ 1 *f b*ð Þ ,*c*

1<*h x*ð Þ<

**Proof.** This follows from taking *b* ¼ *x=*2 and *c* ¼ ð Þ *x* þ 1 *=*2 in Eq. (10), then

**Proof.** Take *b* ¼ *x=*2 and *c* ¼ ð Þ *x* þ 1 *=*2 in Eq. (30). Then the sequence *Sn* of

*N*

1

<sup>2</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>x</sup>* � <sup>1</sup>

2*n* þ 1 þ *x*

� �

2*n* þ 1 þ *x*

2*n* þ 1 þ *x*

*:* (36)

(37)

*n*¼1 *un*

1

<sup>2</sup>*<sup>n</sup>* <sup>þ</sup> *<sup>x</sup>* � <sup>1</sup>

1

<sup>2</sup>*<sup>M</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* � <sup>1</sup>

� �

<sup>2</sup>*<sup>n</sup>* � <sup>1</sup> <sup>þ</sup> *<sup>x</sup>* � <sup>1</sup>

� �

2*N* þ 1 þ *x*

*c* þ 1 *b* þ 1 � �<sup>2</sup>

*Un*ð Þ *an* � *an*þ<sup>1</sup> *:* (31)

*<sup>b</sup>* <sup>þ</sup> <sup>1</sup> (33)

*h c*ð Þ (34)

*:* (35)

(32)

$$\begin{split} \left| S\_{N}^{(k+1)} - S\_{M}^{(k+1)} \right| &\leq k! \sum\_{n=M+1}^{N} \left( \frac{1}{(2n+\varkappa)^{k+1}} - \frac{1}{(2n+1+\varkappa)^{k+1}} \right) \\ &\leq k! \sum\_{n=M+1}^{N} \left( \frac{1}{(2n-1+\varkappa)^{k+1}} - \frac{1}{(2n+1+\varkappa)^{k+1}} \right) \\ &= \frac{k!}{\left(2M+1+\varkappa\right)^{k+1}} - \frac{k!}{\left(2N+1+\varkappa\right)^{k+1}} \\ &< \frac{k!}{\left(2M-1\right)^{k+1}} \to 0 \end{split} \tag{39}$$

as *<sup>M</sup>* ! <sup>∞</sup>, for any *<sup>x</sup>*∈ð Þ �2, <sup>∞</sup> and *<sup>N</sup>* <sup>&</sup>gt; *<sup>M</sup>*. Hence *<sup>S</sup>*ð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>n</sup>* converges uniformly on ð Þ �2, <sup>∞</sup> , i.e., *<sup>h</sup>*ð Þ*<sup>k</sup>* is differentiable on ð Þ �2, <sup>∞</sup> .

Therefore, by induction, *h* has derivatives of all orders on ð Þ �2, ∞ . **Theorem 1.6** Let *a*≥0. Then

$$\log h(\mathbf{x}) = \log h(a) + \sum\_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} \left( \sum\_{n=2}^{\infty} \frac{u\_n}{(n+a)^k} \right) (\mathbf{x} - a)^k \tag{40}$$

for *x*∈ ½ � *a* � 1, *a* þ 1 . **Proof.** Let *H x*ð Þ¼ log *h x*ð Þ. By Theorem 1.5,

$$H^{(k+1)}(\mathbf{x}) = (-\mathbf{1})^k k! \sum\_{n=2}^{\infty} \frac{u\_n}{(n+\mathbf{x})^{k+1}}.\tag{41}$$

Hence

$$|H^{(k+1)}(\mathbf{x})| \le k! \sum\_{n=2}^{\infty} \frac{1}{|n+\mathbf{x}|^{k+1}} \le k! \sum\_{n=2}^{\infty} \frac{1}{(n+a-1)^{k+1}}\tag{42}$$

for *x*∈ ½ � *a* � 1, *a* þ 1 . So by Taylor's inequality, the remainder for the Taylor polynomial for *H x*ð Þ of degree *k* is absolutely bounded above by

$$\frac{1}{k+1} \left( \sum\_{n=2}^{\infty} \frac{1}{(n+a-1)^{k+1}} \right) |\mathbf{x} - a|^{k+1} \tag{43}$$

which tends to 0 as *k* ! ∞, since *a*≥0 and ∣*x* � *a*∣ ≤1. Therefore *H x*ð Þ equals its Taylor expansion about *a* for *x* in the given range.

### **5. Further remarks on** *h*ð Þ **0**

As mentioned in Section 1, not much is known about the quantity *h*ð Þ 0 ≈1*:*62816. We give the following explanation as to why *h*ð Þ 0 might behave specially in a sense.

Note that the only way nontrivial cancelation occurs in Eq. (11) is when *b* ¼ 0. Likewise, nontrivial cancelation occurs in Eq. (10) or property 3 in Section 2 only for ð Þ¼ *b*,*c* ð Þ 0, 1*=*2 and 1ð Þ *=*2, 0 . That is, the victim of any such cancelation is always *<sup>h</sup>*ð Þ <sup>0</sup> or *<sup>h</sup>*ð Þ <sup>0</sup> �<sup>1</sup> . So we must look for other ways to study *h*ð Þ 0 .

Using the two known values *<sup>h</sup>*ð Þ¼ <sup>1</sup>*=*<sup>2</sup> <sup>3</sup>*=*2 and *<sup>h</sup>*ð Þ¼ <sup>1</sup> ffiffi 2 <sup>p</sup> , the following expressions for *h*ð Þ 0 are obtained from Theorem 1.6 by choosing various values for *x* and *a*.

• *x* ¼ 0 and *a* ¼ 1:

$$h(\mathbf{0}) = \sqrt{2} \exp\left(-\sum\_{k=1}^{\infty} \frac{1}{k} \sum\_{n=2}^{\infty} \frac{u\_n}{(n+1)^k}\right) \tag{44}$$

• *x* ¼ 1 and *a* ¼ 0:

$$h(\mathbf{0}) = \sqrt{2} \exp\left(\sum\_{k=1}^{\infty} \frac{(-1)^k}{k} \sum\_{n=2}^{\infty} \frac{\mu\_n}{n^k}\right) \tag{45}$$

• *x* ¼ 0 and *a* ¼ 1*=*2:

$$h(0) = \frac{3}{2} \exp\left(\sum\_{k=1}^{\infty} \frac{1}{k} \sum\_{n=2}^{\infty} \frac{u\_{2n+1}}{(2n+1)^k}\right) \tag{46}$$

$$\bullet \text{ : } \mathbf{x} = \mathbf{1}/2 \text{ and } a = \mathbf{0};$$

$$h(\mathbf{0}) = \frac{3}{2} \exp\left(\sum\_{k=1}^{\infty} \frac{(-1)^k}{k} \sum\_{n=2}^{\infty} \frac{\mu\_{2n}}{\left(2n\right)^k}\right) \tag{47}$$

**Author details**

*Digit Sums and Infinite Products*

*DOI: http://dx.doi.org/10.5772/intechopen.92365*

University of Waterloo, Waterloo, Canada

provided the original work is properly cited.

\*Address all correspondence to: sriasat@uwaterloo.ca

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Samin Riasat

**35**

The Dirichlet series appearing in the above expressions were studied in [9]. We think that these identities and the results from Section 4 might help in shedding some light on the nature of *h*ð Þ 0 .

#### **6. Conclusions and future developments**

We evaluated infinite products involving the digit sum function *sb*ð Þ *n* by splitting the product based on the congruence classes modulo *b*. We illustrated two approaches for doing so, one by direct computation and another using functional equations. For *b* ¼ 2 we proved some analytical results to aid us in understanding the behavior of these products. Many open questions still remain.

Although we only considered the base *b* ¼ 2, many of the results above easily generalize to other bases. One possible direction toward a generalization is to take *un* ¼ �ð Þ<sup>1</sup> *sb*ð Þ *<sup>n</sup>* . Another is *un* <sup>¼</sup> *<sup>ω</sup>sb*ð Þ *<sup>n</sup> <sup>b</sup>* , where *ω<sup>b</sup>* is a primitive *b*-th root of unity. We leave these as work to be done in the future.

*Digit Sums and Infinite Products DOI: http://dx.doi.org/10.5772/intechopen.92365*

Note that the only way nontrivial cancelation occurs in Eq. (11) is when *b* ¼ 0. Likewise, nontrivial cancelation occurs in Eq. (10) or property 3 in Section 2 only for ð Þ¼ *b*,*c* ð Þ 0, 1*=*2 and 1ð Þ *=*2, 0 . That is, the victim of any such cancelation is

expressions for *h*ð Þ 0 are obtained from Theorem 1.6 by choosing various values for

X∞ *k*¼1

*k*¼1

1 *k* X∞ *n*¼2

ð Þ �<sup>1</sup> *<sup>k</sup> k*

X∞ *n*¼2

!

*u*2*<sup>n</sup>* ð Þ 2*n k*

*<sup>b</sup>* , where *ω<sup>b</sup>* is a primitive *b*-th root of unity. We

1 *k* X∞ *n*¼2

!

ð Þ �<sup>1</sup> *<sup>k</sup> k*

!

!

X∞ *n*¼2

*u*2*n*þ<sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>k</sup>*

*un nk*

Using the two known values *<sup>h</sup>*ð Þ¼ <sup>1</sup>*=*<sup>2</sup> <sup>3</sup>*=*2 and *<sup>h</sup>*ð Þ¼ <sup>1</sup> ffiffi

*<sup>h</sup>*ð Þ¼ <sup>0</sup> ffiffi

*<sup>h</sup>*ð Þ¼ <sup>0</sup> ffiffi

*<sup>h</sup>*ð Þ¼ <sup>0</sup> <sup>3</sup>

*<sup>h</sup>*ð Þ¼ <sup>0</sup> <sup>3</sup>

2 <sup>p</sup> exp <sup>X</sup><sup>∞</sup>

<sup>2</sup> exp <sup>X</sup><sup>∞</sup>

<sup>2</sup> exp <sup>X</sup><sup>∞</sup>

*k*¼1

*k*¼1

The Dirichlet series appearing in the above expressions were studied in [9]. We think that these identities and the results from Section 4 might help in shedding

We evaluated infinite products involving the digit sum function *sb*ð Þ *n* by splitting the product based on the congruence classes modulo *b*. We illustrated two approaches for doing so, one by direct computation and another using functional equations. For *b* ¼ 2 we proved some analytical results to aid us in understanding

Although we only considered the base *b* ¼ 2, many of the results above easily generalize to other bases. One possible direction toward a generalization is to take

the behavior of these products. Many open questions still remain.

2 <sup>p</sup> exp �

. So we must look for other ways to study *h*ð Þ 0 .

2

*un* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>k</sup>*

<sup>p</sup> , the following

(44)

(45)

(46)

(47)

always *<sup>h</sup>*ð Þ <sup>0</sup> or *<sup>h</sup>*ð Þ <sup>0</sup> �<sup>1</sup>

*Number Theory and Its Applications*

• *x* ¼ 0 and *a* ¼ 1:

• *x* ¼ 1 and *a* ¼ 0:

• *x* ¼ 0 and *a* ¼ 1*=*2:

• *x* ¼ 1*=*2 and *a* ¼ 0:

some light on the nature of *h*ð Þ 0 .

*un* ¼ �ð Þ<sup>1</sup> *sb*ð Þ *<sup>n</sup>* . Another is *un* <sup>¼</sup> *<sup>ω</sup>sb*ð Þ *<sup>n</sup>*

**34**

leave these as work to be done in the future.

**6. Conclusions and future developments**

*x* and *a*.

### **Author details**

Samin Riasat University of Waterloo, Waterloo, Canada

\*Address all correspondence to: sriasat@uwaterloo.ca

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Robbins D. Solution to problem E 2692. American Mathematical Monthly. 1979;**86**:394-395

[2] Woods DR. Elementary problem proposal E 2692. American Mathematical Monthly. 1978;**85**:48

[3] Allouche J-P, Shallit J. Infinite products associated with counting blocks in binary strings. Journal of the London Mathematical Society. 1989;**39**: 193-204

[4] Allouche J-P, Sondow J. Infinite products with strongly *B*-multiplicative exponents. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Biologica/Errata: Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Biologica. 2008/2010;**28/32**:35-53/253

[5] Allouche J-P. Thue, combinatorics on words, and conjectures inspired by the Thue-Morse sequence. Journal de Théorie des Nombres de Bordeaux. 2015;**27**(2):375-388

[6] Allouche J-P, Shallit J. The ubiquitous Prouhet-Thue-Morse sequence. Sequences and their Applications. In: Ding C, Helleseth T, Niederreiter H, editors. Proceedings of SETA'98. Springer Verlag. 1999; 1-16

[7] Allouche J-P, Riasat S, Shallit J. More infinite products: Thue-Morse and the gamma function. The Ramanujan Journal. 2019;**49**:115-128. DOI: 10.1007/ s11139-017-9981-7

[8] Riasat S. Infinite products involving binary digit sums. In: Kilgour D, Kunze H, Makarov R, Melnik R, Wang X, editors. Recent Advances in Mathematical and Statistical Methods. AMMCS 2017. Springer Proceedings in Mathematics & Statistics. Vol. 259. Cham: Springer; 2017

[9] Allouche J-P, Cohen H. Dirichlet series and curious infinite products. The Bulletin of the London Mathematical Society. 1985;**17**:531-538

**Chapter 3**

Type Hamlet?)

*Anant P. Godbole*

**Abstract**

**1. Introduction**

(∞, Ð

**37**

The Borel-Cantelli Lemmas,

and Their Relationship to Limit

Superior and Limit Inferior of

Sets (or, Can a Monkey Really

The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare's *Hamlet* and any other works one may wish to add to the list will *each* be typed, not once, not twice, but *infinitely often* with a probability of 1. This dramatic fact is a simple consequence of the *Borel-Cantelli lemma* and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of

**Keywords:** Borel-Cantelli lemma, limit superior of sets, limit imferior of sets,

Consider a monkey named Sue who is given a word processor with *N* symbols. We shall assume that these symbols include the 26 letters of the English alphabet (upper and lower case), all the Greek letters, the numbers 0 through 9, a blank space, all the standard punctuation marks (,.; � etc.), and mathematical symbols

, ), ∇, etc.); imagine, in fact, that *N* is so large that the keyboard is capable of typing just anything we might fancy, in any language. (A LATEX editor could do

If Sue is handed such a machine and pounds away, randomly, it is clear that most of what she types will be complete gibberish. A stray segment of sense such as "dog eat HAy" or even "Ronnie ReAGan our 40th PresiD*εητ* of USA" would surprise no one, but how often, we ask, would she successfully manage to type the Constitution of the United States, or Shakespeare's *Hamlet*, or the fundamental mathematical

The purpose of this chapter is to show that if Sue types infinitely, the above works (and any others that one may choose to add to the list) will *each* be typed, not once, not twice, but *infinitely often* with a probability of 1. This dramatic fact is a

limit superior and limit inferior of a sequence of sets.

independence, limit theorems of probability

much of that too, but not in all languages!)

works of the 2026 Fields Medalist(s)?

#### **Chapter 3**

**References**

1979;**86**:394-395

193-204

[1] Robbins D. Solution to problem E 2692. American Mathematical Monthly.

*Number Theory and Its Applications*

[9] Allouche J-P, Cohen H. Dirichlet series and curious infinite products. The Bulletin of the London Mathematical

Society. 1985;**17**:531-538

[2] Woods DR. Elementary problem

Mathematical Monthly. 1978;**85**:48

[3] Allouche J-P, Shallit J. Infinite products associated with counting blocks in binary strings. Journal of the London Mathematical Society. 1989;**39**:

[4] Allouche J-P, Sondow J. Infinite products with strongly *B*-multiplicative exponents. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Biologica/Errata: Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Biologica. 2008/2010;**28/32**:35-53/253

[5] Allouche J-P. Thue, combinatorics on words, and conjectures inspired by the Thue-Morse sequence. Journal de Théorie des Nombres de Bordeaux.

[6] Allouche J-P, Shallit J. The ubiquitous

[7] Allouche J-P, Riasat S, Shallit J. More infinite products: Thue-Morse and the gamma function. The Ramanujan Journal. 2019;**49**:115-128. DOI: 10.1007/

[8] Riasat S. Infinite products involving binary digit sums. In: Kilgour D, Kunze H, Makarov R, Melnik R, Wang X, editors. Recent Advances in Mathematical and Statistical Methods. AMMCS 2017. Springer Proceedings in Mathematics & Statistics. Vol. 259.

Prouhet-Thue-Morse sequence. Sequences and their Applications. In: Ding C, Helleseth T, Niederreiter H, editors. Proceedings of SETA'98. Springer Verlag. 1999; 1-16

2015;**27**(2):375-388

s11139-017-9981-7

Cham: Springer; 2017

**36**

proposal E 2692. American

The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets (or, Can a Monkey Really Type Hamlet?)

*Anant P. Godbole*

#### **Abstract**

The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare's *Hamlet* and any other works one may wish to add to the list will *each* be typed, not once, not twice, but *infinitely often* with a probability of 1. This dramatic fact is a simple consequence of the *Borel-Cantelli lemma* and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of limit superior and limit inferior of a sequence of sets.

**Keywords:** Borel-Cantelli lemma, limit superior of sets, limit imferior of sets, independence, limit theorems of probability

#### **1. Introduction**

Consider a monkey named Sue who is given a word processor with *N* symbols. We shall assume that these symbols include the 26 letters of the English alphabet (upper and lower case), all the Greek letters, the numbers 0 through 9, a blank space, all the standard punctuation marks (,.; � etc.), and mathematical symbols (∞, Ð , ), ∇, etc.); imagine, in fact, that *N* is so large that the keyboard is capable of typing just anything we might fancy, in any language. (A LATEX editor could do much of that too, but not in all languages!)

If Sue is handed such a machine and pounds away, randomly, it is clear that most of what she types will be complete gibberish. A stray segment of sense such as "dog eat HAy" or even "Ronnie ReAGan our 40th PresiD*εητ* of USA" would surprise no one, but how often, we ask, would she successfully manage to type the Constitution of the United States, or Shakespeare's *Hamlet*, or the fundamental mathematical works of the 2026 Fields Medalist(s)?

The purpose of this chapter is to show that if Sue types infinitely, the above works (and any others that one may choose to add to the list) will *each* be typed, not once, not twice, but *infinitely often* with a probability of 1. This dramatic fact is a

simple consequence of the *Borel-Cantelli lemma* and will come as no surprise to anyone who has taken a serious graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence.

hand, and of real numbers, on the other (recall that lim inf *<sup>n</sup>*!<sup>∞</sup> *an* <sup>≤</sup> lim sup*n*!<sup>∞</sup>*an*).

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets…*

�∞ ≤ lim inf *an* ≤ lim sup*an* ≤ ∞,

*ϕ*⊂ lim *An* ⊂ lim *An* ⊂ Ω*:*

ð Þ lim *An <sup>c</sup>* <sup>¼</sup> lim *Ac*

where *Ac* denotes the complement of the set A. Here is an informal proof of the

A few examples should help familiarize the reader with the above notions: the

lim *An* <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* : *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>&</sup>lt;<sup>1</sup>

lim *An* <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* : *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>≤</sup> <sup>1</sup> nf g ð Þ 0, 1 , 0, ð Þ �<sup>1</sup> *:*

In what follows, the set Ω will be taken to be the *sample space* (or set of possible realizations) of a random experiment (one whose outcome cannot be predicted in advance). We shall assume that each subset of Ω that we encounter is *measurable*. In other words, each set *A* will be assumed to belong to the *sigma algebra* A of *events*,

*<sup>n</sup>*, 1 and *<sup>A</sup>*2*<sup>n</sup>* ¼ �1, <sup>1</sup>

*<sup>n</sup>*¼<sup>1</sup>*An.*

*n*

define to be the common value of f g *An* a*:*b*:*f*:*o and f g *An* i*:*o*:* and denote by

lim *An <sup>c</sup>*

If f g *An* a*:*b*:*f*:*o ¼ f g *An* i*:*o*:* , the sequence f g *An* is said to have a limit, which we

<sup>¼</sup> lim *Ac n*,

*n*

*<sup>n</sup>* iff *ω* belongs to all but finitely many *Ac*

*<sup>c</sup>*

*,* then lim *An* <sup>¼</sup> f g<sup>0</sup> and

*<sup>n</sup>* , 0 *,* then

.

*<sup>n</sup>*¼<sup>1</sup>*An.* Likewise, if

*<sup>n</sup>*'s iff *ω*∈

Likewise lim inf *<sup>n</sup>*!<sup>∞</sup>*an* and lim sup*n*!<sup>∞</sup>*an* must both exist, that is,

lim *<sup>n</sup>*!<sup>∞</sup>*An* (note, again, the analogy with real sequences). A useful dual relation between these two sets is

just a finite number of the *An*'s iff *ω* ∉ lim *An* iff *ω*∈ lim *An*

*Example 1*. If *<sup>A</sup>*<sup>1</sup> <sup>⊂</sup> *<sup>A</sup>*<sup>2</sup> <sup>⊂</sup> … , then lim *An* <sup>¼</sup> lim *An* <sup>¼</sup> <sup>∪</sup><sup>∞</sup>

*Example 3*. if *An* is the unit circle with center at ð Þ �<sup>1</sup> *<sup>n</sup>*

which is a class of subsets of Ω satisfying the conditions

*<sup>n</sup>*¼<sup>1</sup>*An* ∈ A.

*P A*f g *<sup>n</sup>* a*:*b*:*f*:*o *:* We next move on to a key concept in probability:

This restriction ensures that sets such as lim *An* and lim *An* are themselves measurable, so that we may meaningfully talk of their probabilities *P A*f g *<sup>n</sup>* i*:*o*:* and

as must lim *An* and lim *An*, with

*DOI: http://dx.doi.org/10.5772/intechopen.93121*

first of these two facts: *ω*∈ lim *A<sup>c</sup>*

*Example 2*. If *<sup>A</sup>*2*n*�<sup>1</sup> ¼ � <sup>1</sup>

• If *<sup>A</sup>* <sup>∈</sup> <sup>A</sup>, then *Ac* <sup>∈</sup> <sup>A</sup> and

• If *<sup>A</sup>*1, *<sup>A</sup>*2, … <sup>∈</sup> <sup>A</sup>, then <sup>∪</sup><sup>∞</sup>

lim *An* ¼ �ð � 1, 1 *.*

and

• *ϕ*∈ A

**39**

second and the third are taken from [2]:

*<sup>A</sup>*<sup>1</sup> <sup>⊃</sup> *<sup>A</sup>*<sup>2</sup> <sup>⊃</sup> *<sup>A</sup>*<sup>3</sup> <sup>⊃</sup> … *,* then lim *An* <sup>¼</sup> lim *An* <sup>¼</sup> <sup>∩</sup><sup>∞</sup>

and

The reader is invited, while reading this chapter, to let his/her imagination run wild, and concoct a plethora of similar examples. A somewhat mundane objection may be raised immediately: how can Sue (or anyone else for that matter) type indefinitely? We shall not dwell on this nonmathematical problem, but will remark instead (and prove a little later) that Sue's never-ending assignment is mathematically equivalent to the task of randomly selecting a number from the interval 0, 1 ½ �.

We would like to mention that our problem is related to the famous "Problem of a printed line," a popular account of which can be found in George Gamow's classic book [1]. The solution presented there, however, is entirely deterministic and of a finite character: the automatic printing press considered by Gamow does not print indefinitely, and the probabilities of various outcomes are not calculated.

#### **2. Limit superior and limit inferior of a sequence of sets**

Consider any sequence f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> of subsets of a set Ω. Points of Ω will be denoted by *ω*. We know that

$$\bigcup\_{n=1}^{\infty} A\_n = \{ a \in \Omega \, : \, a \in A\_n \text{ for some } n \} $$

and

$$\bigcap\_{n=1}^{\infty} A\_n = \{ a \in \Omega \, : \, a \in A\_n \text{ for each } n \}$$

It follows that

$$\bigcap\_{n=1}^{\infty} \stackrel{\Leftrightarrow}{\cup} A\_k = \{ \omega \in \Omega \, : \, \forall n \exists k \ge n \text{ with } \, \omega \in A\_k \} $$

To better understand this somewhat complicated set, we first let *n* ¼ 1 and note that *ω*∈ *Ak* for some *k*≥1, say *k* ¼ *k*1. Letting *n* ¼ *k*1, we see that *ω* must belong to some *Ak*<sup>2</sup> , where *k*<sup>2</sup> ≥*k*1. Continuing in this fashion, we see that *ω* ∈ ∩<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>∪<sup>∞</sup> *<sup>k</sup>*¼*<sup>n</sup>Ak* if and only if *ω*∈ *Ak* for infinitely many *k*'s.

The set ∩<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>∪<sup>∞</sup> *<sup>k</sup>*¼*<sup>n</sup>Ak* is called the *limit superior* of the sequence f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>, and is denoted by lim sup *An*, or, lim *An*, or, rather appropriately, by f g *An* i*:*o*:* , where i.o. stands for "infinitely often."

In a similar fashion, we observe that the set

$$\bigcup\_{n=1}^{\infty} \bigcap\_{k=n}^{\infty} A\_k = \{ \phi \in \Omega : \exists n \text{ such that } \phi \in A\_k \text{ } \forall k \ge n \}$$

is a collection of those points *ω* that belong to *all but a finite number* of the *An*'s. ∪<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>∩<sup>∞</sup> *<sup>k</sup>*¼*<sup>n</sup>Ak* is called the *limit inferior* of the sequence f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> and is usually denoted by lim inf *An* or lim *An*. We prefer the notation f g *An* a*:*b*:*f*:*o (a.b.f.o means "all but finitely often"). Elementary symbol manipulation may be used to prove that lim *An* ⊂ lim *An*. It is easier to note, however, that if *ω* belongs to all but finitely many *An*'s it must necessarily belong to an infinite number of them. The above fact is just one of the many similarities between lim sups and lim infs of sets, on the one *The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets… DOI: http://dx.doi.org/10.5772/intechopen.93121*

hand, and of real numbers, on the other (recall that lim inf *<sup>n</sup>*!<sup>∞</sup> *an* <sup>≤</sup> lim sup*n*!<sup>∞</sup>*an*). Likewise lim inf *<sup>n</sup>*!<sup>∞</sup>*an* and lim sup*n*!<sup>∞</sup>*an* must both exist, that is,

�∞ ≤ lim inf *an* ≤ lim sup*an* ≤ ∞,

as must lim *An* and lim *An*, with

$$
\phi \subset \underline{\lim} A\_n \subset \overline{\lim} A\_n \subset \Omega.
$$

If f g *An* a*:*b*:*f*:*o ¼ f g *An* i*:*o*:* , the sequence f g *An* is said to have a limit, which we define to be the common value of f g *An* a*:*b*:*f*:*o and f g *An* i*:*o*:* and denote by lim *<sup>n</sup>*!<sup>∞</sup>*An* (note, again, the analogy with real sequences).

A useful dual relation between these two sets is

$$\left(\overline{\lim} A\_n\right)^c = \underline{\lim} A\_n^c,$$

and

simple consequence of the *Borel-Cantelli lemma* and will come as no surprise to anyone who has taken a serious graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary

indefinitely, and the probabilities of various outcomes are not calculated.

**2. Limit superior and limit inferior of a sequence of sets**

The reader is invited, while reading this chapter, to let his/her imagination run wild, and concoct a plethora of similar examples. A somewhat mundane objection may be raised immediately: how can Sue (or anyone else for that matter) type indefinitely? We shall not dwell on this nonmathematical problem, but will remark instead (and prove a little later) that Sue's never-ending assignment is mathematically equivalent to the task of randomly selecting a number from the interval 0, 1 ½ �. We would like to mention that our problem is related to the famous "Problem of a printed line," a popular account of which can be found in George Gamow's classic book [1]. The solution presented there, however, is entirely deterministic and of a finite character: the automatic printing press considered by Gamow does not print

*An* ¼ f g *ω* ∈ Ω : *ω*∈ *An* for some *n*

*An* ¼ f g *ω*∈ Ω : *ω*∈ *An* for each *n*

*Ak* ¼ f g *ω*∈ Ω : ∀*n*∃*k*≥*n* with *ω* ∈ *Ak*

To better understand this somewhat complicated set, we first let *n* ¼ 1 and note that *ω*∈ *Ak* for some *k*≥1, say *k* ¼ *k*1. Letting *n* ¼ *k*1, we see that *ω* must belong to

*<sup>k</sup>*¼*<sup>n</sup>Ak* is called the *limit superior* of the sequence f g *An* <sup>∞</sup>

*Ak* ¼ f g *ω* ∈ Ω : ∃*n* such that *ω*∈ *Ak* ∀*k*≥*n*

is a collection of those points *ω* that belong to *all but a finite number* of the *An*'s.

by lim inf *An* or lim *An*. We prefer the notation f g *An* a*:*b*:*f*:*o (a.b.f.o means "all but finitely often"). Elementary symbol manipulation may be used to prove that lim *An* ⊂ lim *An*. It is easier to note, however, that if *ω* belongs to all but finitely many *An*'s it must necessarily belong to an infinite number of them. The above fact is just one of the many similarities between lim sups and lim infs of sets, on the one

denoted by lim sup *An*, or, lim *An*, or, rather appropriately, by f g *An* i*:*o*:* , where i.o.

some *Ak*<sup>2</sup> , where *k*<sup>2</sup> ≥*k*1. Continuing in this fashion, we see that *ω* ∈ ∩<sup>∞</sup>

*<sup>k</sup>*¼*<sup>n</sup>Ak* is called the *limit inferior* of the sequence f g *An* <sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup> of subsets of a set Ω. Points of Ω will be denoted

*<sup>n</sup>*¼<sup>1</sup>∪<sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup> and is usually denoted

*<sup>k</sup>*¼*<sup>n</sup>Ak* if

*<sup>n</sup>*¼<sup>1</sup>, and is

understanding of the concept of independence.

*Number Theory and Its Applications*

Consider any sequence f g *An* <sup>∞</sup>

∪ ∞ *n*¼1

> ∩ ∞ *n*¼1

∩ ∞ *n*¼1 ∪ ∞ *k*¼*n*

and only if *ω*∈ *Ak* for infinitely many *k*'s.

In a similar fashion, we observe that the set

*<sup>n</sup>*¼<sup>1</sup>∪<sup>∞</sup>

∪ ∞ *n*¼1 ∩ ∞ *k*¼*n*

stands for "infinitely often."

by *ω*. We know that

It follows that

The set ∩<sup>∞</sup>

∪<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>∩<sup>∞</sup>

**38**

and

$$(\underline{\lim} \underline{A}\_n)^c = \overline{\lim} \underline{A}\_n^c$$

where *Ac* denotes the complement of the set A. Here is an informal proof of the first of these two facts: *ω*∈ lim *A<sup>c</sup> <sup>n</sup>* iff *ω* belongs to all but finitely many *Ac <sup>n</sup>*'s iff *ω*∈ just a finite number of the *An*'s iff *ω* ∉ lim *An* iff *ω*∈ lim *An <sup>c</sup>* .

A few examples should help familiarize the reader with the above notions: the second and the third are taken from [2]:

*Example 1*. If *<sup>A</sup>*<sup>1</sup> <sup>⊂</sup> *<sup>A</sup>*<sup>2</sup> <sup>⊂</sup> … , then lim *An* <sup>¼</sup> lim *An* <sup>¼</sup> <sup>∪</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*An.* Likewise, if *<sup>A</sup>*<sup>1</sup> <sup>⊃</sup> *<sup>A</sup>*<sup>2</sup> <sup>⊃</sup> *<sup>A</sup>*<sup>3</sup> <sup>⊃</sup> … *,* then lim *An* <sup>¼</sup> lim *An* <sup>¼</sup> <sup>∩</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*An.*

*Example 2*. If *<sup>A</sup>*2*n*�<sup>1</sup> ¼ � <sup>1</sup> *<sup>n</sup>*, 1 and *<sup>A</sup>*2*<sup>n</sup>* ¼ �1, <sup>1</sup> *n ,* then lim *An* <sup>¼</sup> f g<sup>0</sup> and lim *An* ¼ �ð � 1, 1 *.*

*Example 3*. if *An* is the unit circle with center at ð Þ �<sup>1</sup> *<sup>n</sup> <sup>n</sup>* , 0 *,* then

$$\underline{\lim} A\_n = \left\{ (\mathfrak{x}, \mathfrak{y}) : \mathfrak{x}^2 + \mathfrak{y}^2 < 1 \right\}.$$

and

$$\overline{\lim}A\_n = \left\{ (\mathfrak{x}, \mathfrak{y}) : \mathfrak{x}^2 + \mathfrak{y}^2 \le \mathbf{1} \right\} \backslash \left\{ (\mathbf{0}, \mathbf{1}), (\mathbf{0}, -\mathbf{1}) \right\}.$$

In what follows, the set Ω will be taken to be the *sample space* (or set of possible realizations) of a random experiment (one whose outcome cannot be predicted in advance). We shall assume that each subset of Ω that we encounter is *measurable*. In other words, each set *A* will be assumed to belong to the *sigma algebra* A of *events*, which is a class of subsets of Ω satisfying the conditions

$$\bullet \,\,\phi \in \mathcal{A}$$


This restriction ensures that sets such as lim *An* and lim *An* are themselves measurable, so that we may meaningfully talk of their probabilities *P A*f g *<sup>n</sup>* i*:*o*:* and *P A*f g *<sup>n</sup>* a*:*b*:*f*:*o *:* We next move on to a key concept in probability:

**Definition**: The sequence of events f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> will be said to be *independent*, if for each finite subcollection *An*<sup>1</sup> , *An*<sup>2</sup> , … , *Ank* ,

which proves that

proving the result.

*probability of 1.*

sequence f g *An* <sup>∞</sup>

Thus *P* ∩*<sup>n</sup>*

*probability of 1.*

**41**

*<sup>k</sup>*¼<sup>1</sup>*Ck*

*n* ! ∞, and using Lemma 2.2.

often (or never) is zero.)

*P Ac*

*DOI: http://dx.doi.org/10.5772/intechopen.93121*

being typed correctly. It is clear that *P A*ð Þ¼ <sup>1</sup>

proof is complete. (Notice how the events f g *An* <sup>∞</sup>

**Lemma 2.4.** *If P C*<sup>ð</sup> *n) = 1 (n = 1,2,...), then P* <sup>∩</sup><sup>∞</sup>

*P* ∩*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*Ck* � �<sup>≥</sup> <sup>X</sup>*<sup>n</sup>*

this guarantees their independence.)

**Proof:** Boole's inequality states that

the ð Þ *nM* th random strokes. It is evident that *P A*ð Þ¼ *<sup>n</sup>* <sup>1</sup>

*<sup>n</sup>*¼<sup>1</sup> is independent. Since <sup>P</sup><sup>∞</sup>

*<sup>n</sup>* <sup>a</sup>*:*b*:*f*:*<sup>o</sup> � �<sup>≤</sup> <sup>X</sup><sup>∞</sup>

*n*¼1

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets…*

As seen by the dates on Refs. [3, 4], the Borel-Cantelli lemmas are classical, and now part of virtually all graduate level books on Probability such as [2]. Since then, for over 100 years, the literature on the lemmas has focused on weakening the independence requirement in the second lemma, or looking at more complicated probability models that yield the same conclusions. See for example [5–7]. What distinguishes this work from these and others is that we provide a very down-toearth application that forces the reader to come to terms with the notions of independence and infinity, as opposed to the finite samples one has in statistical situations. It is a paper that we feel can cause amusement, astonishment, false disbelief, and, ultimately, understanding. With this backdrop, we are now in a position to start establishing the claim made at the beginning of this chapter:

**Corollary 2.3.** *If Sue types indefinitely, by successively and independently choosing one of the N available keys, then any specific work containing a total of M characters (including all the blanks, of course) will be typed by her indfinitely often, with a*

**Proof.** Let *A*<sup>1</sup> be the event that Sue's first *M* random choices lead to the work

successful completion of the task between the ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> st and 2ð Þ *<sup>M</sup>* th keystrokes. In general, *An* denotes the completion of a flawless job between ð Þ ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> st and

by part (b) of the Borel-Cantelli lemma that *P A*ð Þ¼ *<sup>n</sup>* i*:*o*:* 1. Since the probability that the work is typed correctly infinitely often is at least as large as *P A*ð Þ *<sup>n</sup>* i*:*o*:* , the

*k*¼1

**Corollary 2.5.** *If Sue types indefinitely, then every piece of writing (of any finite*

*individuals; meaningful or gibberish) will be typed by her, infinitely often each, with a*

claimed. (Note: this implies that the probability that *some* work is typed finitely

**Proof.** Denote the works by *B*1, *B*2, *:* … , and set *An* ¼ f g *Bn* i*:*o*:* , ð Þ *n* ¼ 1, 2, … *:*

*Example 4*. (*Statistical tests of hypotheses*) If a fair coin is tossed infinitely often, a sequence of 10<sup>6</sup> consecutive heads will appear infinitely often with probability 1*.*

*length whatsoever; published, unpublished, or yet to be written by yet unborn*

By Lemma 2.2, *P A*ð Þ¼ *<sup>n</sup>* 1 for each *<sup>n</sup>* and thus, by Lemma 2.4, *<sup>P</sup>* <sup>∩</sup><sup>∞</sup>

1 *N<sup>M</sup>*

*<sup>n</sup>*¼<sup>1</sup>*P A*ð Þ¼ *<sup>n</sup>*

*<sup>n</sup>*¼<sup>1</sup>*Cn* � � <sup>¼</sup> <sup>1</sup>*.*

*P C*ð Þ� *<sup>k</sup>* ð Þ *n* � 1 *:*

� � <sup>¼</sup> 1 for each *<sup>n</sup>*. The required conclusion is obtained on letting

� �. Similarly, let *<sup>A</sup>*<sup>2</sup> denote a

� � ð Þ *<sup>n</sup>* <sup>≥</sup><sup>1</sup> and that the

*<sup>n</sup>*¼<sup>1</sup> are defined using disjoint blocks;

*<sup>n</sup>*¼<sup>1</sup>*An* � � <sup>¼</sup> 1, as

� � <sup>¼</sup> <sup>∞</sup>, it follows

*N<sup>M</sup>*

P<sup>∞</sup> *n*¼1 1 *N<sup>M</sup>*

*P* ∩ ∞ *k*¼*n Ac K* � �

¼ 0,

$$P(A\_{n\_1} \cap A\_{n\_2} \cap \dots \cap A\_{n\_k}) = P(A\_{n\_1}) \cdot P(A\_{n\_2}) \cdot \dots \cdot P(A\_{n\_k}) \cdot \alpha$$

Stated informally, this means that the occurrence (or nonoccurrence) of any finite subcollection *An*<sup>1</sup> , *An*<sup>2</sup> , … , *Ank* � � does not affect the probability of occurrence of another disjoint collection *Am*<sup>1</sup> , *Am*<sup>2</sup> , … , *Am*<sup>ℓ</sup> f g.

The events f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> that represent the successive outcomes of an infinite cointossing experiment are usually assumed, on intuitive and empirical grounds, to be independent. We shall make the same assumption regarding Sue's successive choices f g *Bn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> of a keyboard's key.

The *Borel-Cantelli lemma* is a two-pronged theorem, which asserts that the probability of occurrence of an infinite number of the independent events f g *An* <sup>∞</sup> *n*¼1 is zero or one:

**Theorem 2.1.** *(The Borel-Cantelli lemma,* [3, 4]).


The following lemma can be proved using elementary properties of probability measures:

**Lemma 2.2.** *If A*f g*<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *is an increasing (decreasing) sequence of events then* lim *<sup>n</sup>*!∞*P A*ð Þ¼ *<sup>n</sup> <sup>P</sup>* <sup>∪</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*An* � � *P*∩<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*An* � �*: Proof of Theorem 2.1*

a. For any *n*, we note that

$$P(A\_n \text{ i.o.}) \le P(\cup\_{k \ge n} A\_k) \le \sum\_{k=n}^{\infty} P(A\_k).$$

On letting *n* ! ∞, we see that

$$P(A\_n \text{ i.o.}) \le \lim\_{n \to \infty} \sum\_{k=n}^{\infty} P(A\_k) = \mathbf{0},$$

proving part (a).

b. We shall prove that *P Ac <sup>n</sup>* <sup>a</sup>*:*b*:*f*:*<sup>o</sup> � � <sup>¼</sup> 0. Let *<sup>m</sup>*, *n m*ð Þ <sup>≥</sup>*<sup>n</sup>* be arbitrary. Note that

$$P(\bigcap\_{k=n}^{m} A\_k^c) = \prod\_{k=n}^{m} P(A\_k^c) = \prod\_{k=n}^{m} (1 - P(A\_k)) \le \exp\left\{-\sum\_{k=n}^{m} P(A\_k)\right\},$$

where the last inequality follows from the fact that 1 � *<sup>x</sup>*≤*e*�*<sup>x</sup>* ð Þ *<sup>x</sup>*≥<sup>0</sup> . Lemma 2.1 now gives us that

$$\lim\_{m \to \infty} P\left(\bigcap\_{k=n}^m A\_k^{\varepsilon}\right) = P\left(\bigcap\_{k=n}^\infty A\_k^{\varepsilon}\right) = \mathbf{0} \,\,\,\forall n, \varepsilon$$

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets… DOI: http://dx.doi.org/10.5772/intechopen.93121*

which proves that

**Definition**: The sequence of events f g *An* <sup>∞</sup>

*P An*1∩*An*2∩ … ∩*Ank*

of another disjoint collection *Am*<sup>1</sup> , *Am*<sup>2</sup> , … , *Am*<sup>ℓ</sup> f g.

*<sup>n</sup>*¼<sup>1</sup> of a keyboard's key.

*<sup>n</sup>*¼<sup>1</sup>*An* � � *P*∩<sup>∞</sup>

**Theorem 2.1.** *(The Borel-Cantelli lemma,* [3, 4]).

*<sup>n</sup>*¼<sup>1</sup> *is any sequence of events, then* <sup>P</sup><sup>∞</sup>

*<sup>n</sup>*¼<sup>1</sup>*P A*ð Þ*<sup>n</sup> converges or diverges*.

*<sup>n</sup>*¼<sup>1</sup>*An* � �*:*

*P A*ð Þ *<sup>n</sup>* <sup>i</sup>*:*o*:* <sup>≤</sup> lim*<sup>n</sup>*!<sup>∞</sup>

*k*¼*n*

where the last inequality follows from the fact that 1 � *<sup>x</sup>*≤*e*�*<sup>x</sup>* ð Þ *<sup>x</sup>*≥<sup>0</sup> . Lemma

¼ *P* ∩ ∞ *k*¼*n Ac k* � �

*P A*ð Þ *<sup>n</sup>* <sup>i</sup>*:*o*:* <sup>≤</sup>*P*ð Þ <sup>∪</sup>*<sup>k</sup>*≥*nAk* <sup>≤</sup> <sup>X</sup><sup>∞</sup>

X∞ *k*¼*n*

� � <sup>¼</sup> *P An*<sup>1</sup> ð Þ� *P An*<sup>2</sup> ð Þ� … � *P Ank*

Stated informally, this means that the occurrence (or nonoccurrence) of any

tossing experiment are usually assumed, on intuitive and empirical grounds, to be independent. We shall make the same assumption regarding Sue's successive

The *Borel-Cantelli lemma* is a two-pronged theorem, which asserts that the probability of occurrence of an infinite number of the independent events f g *An* <sup>∞</sup>

� � does not affect the probability of occurrence

*<sup>n</sup>*¼<sup>1</sup> that represent the successive outcomes of an infinite coin-

*<sup>n</sup>*¼<sup>1</sup> *is an independence sequence, then P A*ð Þ *<sup>n</sup>* i*:o: equals* 0 *or* 1 *according as*

*<sup>n</sup>*¼<sup>1</sup> *is an increasing (decreasing) sequence of events then*

*k*¼*n*

*P A*ð Þ¼ *<sup>k</sup>* 0,

*<sup>n</sup>* <sup>a</sup>*:*b*:*f*:*<sup>o</sup> � � <sup>¼</sup> 0. Let *<sup>m</sup>*, *n m*ð Þ <sup>≥</sup>*<sup>n</sup>* be arbitrary. Note

¼ 0 ∀*n*,

X*m k*¼*n*

*P A*ð Þ*<sup>k</sup>* ( )

,

ð Þ 1 � *P A*ð Þ*<sup>k</sup>* ≤ exp �

*P A*ð Þ*<sup>k</sup>*

The following lemma can be proved using elementary properties of probability

each finite subcollection *An*<sup>1</sup> , *An*<sup>2</sup> , … , *Ank* ,

*Number Theory and Its Applications*

finite subcollection *An*<sup>1</sup> , *An*<sup>2</sup> , … , *Ank*

The events f g *An* <sup>∞</sup>

choices f g *Bn* <sup>∞</sup>

is zero or one:

a. If f g *An* <sup>∞</sup>

b. *If A*f g*<sup>n</sup>* <sup>∞</sup>

measures:

*P A*ð Þ¼ *<sup>n</sup>* i*:*o*:* 0.

*the series* P<sup>∞</sup>

**Lemma 2.2.** *If A*f g*<sup>n</sup>* <sup>∞</sup>

*Proof of Theorem 2.1*

a. For any *n*, we note that

On letting *n* ! ∞, we see that

b. We shall prove that *P Ac*

*k*¼*n*

*P Ac k* � � <sup>¼</sup> <sup>Y</sup>*<sup>m</sup>*

lim*<sup>m</sup>*!<sup>∞</sup>*<sup>P</sup>* <sup>∩</sup> *m k*¼*n Ac k* � �

lim *<sup>n</sup>*!∞*P A*ð Þ¼ *<sup>n</sup> <sup>P</sup>* <sup>∪</sup><sup>∞</sup>

proving part (a).

*P* ∩*<sup>m</sup> <sup>k</sup>*¼*nAc k* � � <sup>¼</sup> <sup>Y</sup>*<sup>m</sup>*

2.1 now gives us that

**40**

that

*<sup>n</sup>*¼<sup>1</sup> will be said to be *independent*, if for

� �*:*

*<sup>n</sup>*¼<sup>1</sup>*P A*ð Þ*<sup>n</sup>* <sup>&</sup>lt; <sup>∞</sup> *implies that*

*n*¼1

$$P(A\_n^\epsilon \text{ a.b.f.o}) \le \sum\_{n=1}^\infty P\left(\bigcap\_{k=n}^\infty A\_k^\epsilon\right) = \mathbf{0},$$

proving the result.

As seen by the dates on Refs. [3, 4], the Borel-Cantelli lemmas are classical, and now part of virtually all graduate level books on Probability such as [2]. Since then, for over 100 years, the literature on the lemmas has focused on weakening the independence requirement in the second lemma, or looking at more complicated probability models that yield the same conclusions. See for example [5–7]. What distinguishes this work from these and others is that we provide a very down-toearth application that forces the reader to come to terms with the notions of independence and infinity, as opposed to the finite samples one has in statistical situations. It is a paper that we feel can cause amusement, astonishment, false disbelief, and, ultimately, understanding. With this backdrop, we are now in a position to start establishing the claim made at the beginning of this chapter:

**Corollary 2.3.** *If Sue types indefinitely, by successively and independently choosing one of the N available keys, then any specific work containing a total of M characters (including all the blanks, of course) will be typed by her indfinitely often, with a probability of 1.*

**Proof.** Let *A*<sup>1</sup> be the event that Sue's first *M* random choices lead to the work being typed correctly. It is clear that *P A*ð Þ¼ <sup>1</sup> 1 *N<sup>M</sup>* � �. Similarly, let *<sup>A</sup>*<sup>2</sup> denote a successful completion of the task between the ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> st and 2ð Þ *<sup>M</sup>* th keystrokes. In general, *An* denotes the completion of a flawless job between ð Þ ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> st and the ð Þ *nM* th random strokes. It is evident that *P A*ð Þ¼ *<sup>n</sup>* <sup>1</sup> *N<sup>M</sup>* � � ð Þ *<sup>n</sup>* <sup>≥</sup><sup>1</sup> and that the sequence f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> is independent. Since <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*P A*ð Þ¼ *<sup>n</sup>* P<sup>∞</sup> *n*¼1 1 *N<sup>M</sup>* � � <sup>¼</sup> <sup>∞</sup>, it follows by part (b) of the Borel-Cantelli lemma that *P A*ð Þ¼ *<sup>n</sup>* i*:*o*:* 1. Since the probability that the work is typed correctly infinitely often is at least as large as *P A*ð Þ *<sup>n</sup>* i*:*o*:* , the proof is complete. (Notice how the events f g *An* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> are defined using disjoint blocks; this guarantees their independence.)

**Lemma 2.4.** *If P C*<sup>ð</sup> *n) = 1 (n = 1,2,...), then P* <sup>∩</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*Cn* � � <sup>¼</sup> <sup>1</sup>*.* **Proof:** Boole's inequality states that

$$P(\bigcap\_{k=1}^{n} \mathbf{C}\_{k}) \ge \sum\_{k=1}^{n} P(\mathbf{C}\_{k}) - (n - 1).$$

Thus *P* ∩*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*Ck* � � <sup>¼</sup> 1 for each *<sup>n</sup>*. The required conclusion is obtained on letting *n* ! ∞, and using Lemma 2.2.

**Corollary 2.5.** *If Sue types indefinitely, then every piece of writing (of any finite length whatsoever; published, unpublished, or yet to be written by yet unborn individuals; meaningful or gibberish) will be typed by her, infinitely often each, with a probability of 1.*

**Proof.** Denote the works by *B*1, *B*2, *:* … , and set *An* ¼ f g *Bn* i*:*o*:* , ð Þ *n* ¼ 1, 2, … *:* By Lemma 2.2, *P A*ð Þ¼ *<sup>n</sup>* 1 for each *<sup>n</sup>* and thus, by Lemma 2.4, *<sup>P</sup>* <sup>∩</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*An* � � <sup>¼</sup> 1, as claimed. (Note: this implies that the probability that *some* work is typed finitely often (or never) is zero.)

*Example 4*. (*Statistical tests of hypotheses*) If a fair coin is tossed infinitely often, a sequence of 10<sup>6</sup> consecutive heads will appear infinitely often with probability 1*.*

Now, if a coin (of unknown origin) were tossed a million times, and a head appeared each time, the "null" statistical hypothesis

$$H\_0: \text{The coin is fair } \left(p = 1/2\right)$$

A typical element of Ω might be *ω* ¼ 0010011 … . It is well known (and easily proved) that Ω is an uncountable set. It seems reasonable, then, to assign probability

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets…*

The next step is crucial. We identify each element of Ω with the real number in the interval [0,1] that has the same binary expansion. For example, the sample outcome THTHTH ... is identified with the real number 0.01010101 ... which equals

1

A problem arises immediately: Numbers of the form *k=*2*n*, where *k* and *n* are positive integers, do not have a unique binary representation. In other words, two different sample outcomes such as HTTTTT... and THHHH... would correspond to the same real number 1/2 (since 1/2 = 0.0111... = 0.100...), and the correspondence between Ω and [0,1] would not, consequently, be one-to-one. We note, however, that numbers of the form *k=*2*<sup>n</sup>* constitute a denumerable set, and that there are two sample outcomes that correspond to each such number. If one, but not both, of each of these outcomes were to be removed from Ω, we would be left with a one-to-one map from a censored sample space Ω<sup>0</sup> onto [0,1]. Moreover, our assumption regarding individual sample points forces *P* ΩnΩ<sup>0</sup> ð Þ to equal zero. Thus, if a set of zero probability is thrown out from the original sample space, we may let Ω ¼ ½ � 0, 1 and derive great satisfaction from the knowledge that this would not change the

It is possible to show, in a somewhat non-rigorous fashion (i.e., without using much measure theory), or rigorously, by introducing Lebesgue measure, that infinite coin tossing is mathematically equivalent to choosing a number randomly from the interval [0,1]. It can be shown, in a completely analogous way, that infinite random typewriting is equivalent to the single random choice of a number in [0,1]. We need of course, to consider the *N*-ary representation of numbers in [0,1], instead of their binary expansion (where *N* ithe number of typewriter keys).

*Example 5.* (*Random Numbers*) Let the random variable *X* denote the random

where *rn* is the *n*'th rational. Since, *P X*ð Þ¼ ¼ *rn* 0 for each *n*, we have that

*P X*ð is rationalÞ ¼ 0*:*

*P X*ð is rationalÞ ¼ 1!

the number of times *j* appears in the first *n* digits of its decimal expansion *n* � �

[3]: A number in [0,1] is said to be *normal*, if its decimal representation has,

We would like to next state a thrilling result, called *Borel's law of normal numbers*

This result may be compared with a mundane fact of "reality": If a person, computer, pointer, or random number generator were asked to choose *X*, limitations of measurement accuracy (or decimal point restrictions) would systematically

*n*¼1

*P X*ð Þ ¼ *rn*,

¼ 1 10

*P X*<sup>ð</sup> is rationalÞ ¼ <sup>X</sup><sup>∞</sup>

exclude irrational X's, leading to the "conclusion" that

asymptotically, an equal frequency of the digits 0 through 9:

<sup>24</sup> <sup>þ</sup> … <sup>¼</sup> <sup>1</sup>

3 *:*

0 21 <sup>þ</sup> 1 22 <sup>þ</sup> 0 23 <sup>þ</sup>

answer to any of our probability calculations.

However, we shall not do so here.

lim*<sup>n</sup>*!<sup>∞</sup>

**43**

choice of a number from [0,1]. Then

zero to each sample point.

*DOI: http://dx.doi.org/10.5772/intechopen.93121*

would be summarily rejected at most conventional (5, 1, 0.00001%) levels of significance. The point to note, however, is that such "extreme" and "erratic" behavior *will* be exhibited on an infinite number of occasions by any fair coin (and by all coins with *P H*ð Þ>0Þ, with a probability of 1.

Similarly, if a fair coin is tossed infinitely often, an *n*-long alternating sequence *HTHT* … *HT* (*n* is arbitrary) will appear infinitely often, almost certainly. This fact may be compared with the conclusion of a standard nonparametric statistical procedure, *the run test*: the fair coin hypothesis would be vigorously rejected, using this test, if a large number of coin tosses yielded an alternating sequence of heads and tails.

#### **3. A probability model for infinite coin tossing**

In the above discussion, we often concluded that a particular event (e.g., *Hamlet* is typed infinitely often) occurred with probability 1. One fundamental question that we did not address, however, was the following: just what probability model describes infinite coin tossing or simian typewriting? Put another way, what are the sample spaces associated with these two experiments? And what exactly is the probability of an event defined to be? We realize then, in retrospect, that we had put the cart before the horse; various events were shown to have probability 1, by *assuming* the existence of a logically consistent probability (measure) on a sample space that had not been fully described. This practice is fairly standard in the teaching of probability; for example, sequences f g *Xn* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> of independent and identically distributed (i.i.d.) random variables are often introduced as mathematical objects before their existence is proved, using Kolmogorov's famous Consistency Theorem. Such an approach is often beneficial; as Billingsley [8] wrote, "It is instructive... to see the show in rehearsal as well as in performance."

We shall start by noting that three tosses of a fair coin lead to the eight-point sample space

$$\textsf{\mathfrak{Q}} = \{\textsf{HHH}, \textsf{HHT}, \textsf{HTH}, \textsf{HTT}, \textsf{THH}, \textsf{THT}, \textsf{TTH}, \textsf{TTT}\}$$

It seems reasonable to assign probability 1/8 to each of these eight points; thus the probability *P A*ð Þ of *any* subset *A* may be defined by

$$P(A) = \frac{\text{number of points in } A}{8}$$

Our analysis is thus complete, and can easily be extended to any finite number of coin tosses. The situation gets rapidly more complicated if the coin is tossed endlessly. This experiment cannot be conceived, carried out, or justified "in practice," and our neat conclusions would be rendered meaningless if we were unable to mathematically model our procedure. Happily, however, this is not the case. We simply let

> Ω ¼ *ω* : *ω* is an infinite sequence of H<sup>0</sup> s and T<sup>0</sup> f gs ¼ *ω* : *ω* is an infinite sequence of 1<sup>0</sup> s and 0<sup>0</sup> f gs

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets… DOI: http://dx.doi.org/10.5772/intechopen.93121*

A typical element of Ω might be *ω* ¼ 0010011 … . It is well known (and easily proved) that Ω is an uncountable set. It seems reasonable, then, to assign probability zero to each sample point.

The next step is crucial. We identify each element of Ω with the real number in the interval [0,1] that has the same binary expansion. For example, the sample outcome THTHTH ... is identified with the real number 0.01010101 ... which equals

$$\frac{\mathbf{0}}{2^1} + \frac{\mathbf{1}}{2^2} + \frac{\mathbf{0}}{2^3} + \frac{\mathbf{1}}{2^4} + \dots = \frac{\mathbf{1}}{3}.$$

A problem arises immediately: Numbers of the form *k=*2*n*, where *k* and *n* are positive integers, do not have a unique binary representation. In other words, two different sample outcomes such as HTTTTT... and THHHH... would correspond to the same real number 1/2 (since 1/2 = 0.0111... = 0.100...), and the correspondence between Ω and [0,1] would not, consequently, be one-to-one. We note, however, that numbers of the form *k=*2*<sup>n</sup>* constitute a denumerable set, and that there are two sample outcomes that correspond to each such number. If one, but not both, of each of these outcomes were to be removed from Ω, we would be left with a one-to-one map from a censored sample space Ω<sup>0</sup> onto [0,1]. Moreover, our assumption regarding individual sample points forces *P* ΩnΩ<sup>0</sup> ð Þ to equal zero. Thus, if a set of zero probability is thrown out from the original sample space, we may let Ω ¼ ½ � 0, 1 and derive great satisfaction from the knowledge that this would not change the answer to any of our probability calculations.

It is possible to show, in a somewhat non-rigorous fashion (i.e., without using much measure theory), or rigorously, by introducing Lebesgue measure, that infinite coin tossing is mathematically equivalent to choosing a number randomly from the interval [0,1]. It can be shown, in a completely analogous way, that infinite random typewriting is equivalent to the single random choice of a number in [0,1]. We need of course, to consider the *N*-ary representation of numbers in [0,1], instead of their binary expansion (where *N* ithe number of typewriter keys). However, we shall not do so here.

*Example 5.* (*Random Numbers*) Let the random variable *X* denote the random choice of a number from [0,1]. Then

$$P(\mathbf{X} \text{ is rational}) = \sum\_{n=1}^{\infty} P(\mathbf{X} = r\_n, \mathbf{})$$

where *rn* is the *n*'th rational. Since, *P X*ð Þ¼ ¼ *rn* 0 for each *n*, we have that

$$P(X \text{ is rational}) = 0.$$

This result may be compared with a mundane fact of "reality": If a person, computer, pointer, or random number generator were asked to choose *X*, limitations of measurement accuracy (or decimal point restrictions) would systematically exclude irrational X's, leading to the "conclusion" that

$$P(X \text{ is rational}) = 1!$$

We would like to next state a thrilling result, called *Borel's law of normal numbers* [3]: A number in [0,1] is said to be *normal*, if its decimal representation has, asymptotically, an equal frequency of the digits 0 through 9:

$$\lim\_{n \to \infty} \left\{ \frac{\text{the number of times } j \text{ appears in the first } n \text{ digits of its decimal expansion}}{n} \right\} = \frac{1}{10}$$

Now, if a coin (of unknown origin) were tossed a million times, and a head

*H*<sup>0</sup> : The coin is fair ð Þ *p* ¼ 1*=*2

would be summarily rejected at most conventional (5, 1, 0.00001%) levels of significance. The point to note, however, is that such "extreme" and "erratic" behavior *will* be exhibited on an infinite number of occasions by any fair coin (and

Similarly, if a fair coin is tossed infinitely often, an *n*-long alternating sequence *HTHT* … *HT* (*n* is arbitrary) will appear infinitely often, almost certainly. This fact may be compared with the conclusion of a standard nonparametric statistical procedure, *the run test*: the fair coin hypothesis would be vigorously rejected, using this test, if a large number of coin tosses yielded an alternating sequence of heads and

In the above discussion, we often concluded that a particular event (e.g., *Hamlet* is typed infinitely often) occurred with probability 1. One fundamental question that we did not address, however, was the following: just what probability model describes infinite coin tossing or simian typewriting? Put another way, what are the sample spaces associated with these two experiments? And what exactly is the probability of an event defined to be? We realize then, in retrospect, that we had put the cart before the horse; various events were shown to have probability 1, by *assuming* the existence of a logically consistent probability (measure) on a sample space that had not been fully described. This practice is fairly standard in the

cally distributed (i.i.d.) random variables are often introduced as mathematical objects before their existence is proved, using Kolmogorov's famous Consistency Theorem. Such an approach is often beneficial; as Billingsley [8] wrote, "It is

We shall start by noting that three tosses of a fair coin lead to the eight-point

Ω ¼ f g *HHH*, *HHT*, *HTH*, *HTT*, *THH*, *THT*, *TTH*, *TTT*

It seems reasonable to assign probability 1/8 to each of these eight points; thus

*P A*ð Þ¼ number of points in *<sup>A</sup>* 8

Ω ¼ *ω* : *ω* is an infinite sequence of H<sup>0</sup>

¼ *ω* : *ω* is an infinite sequence of 1<sup>0</sup>

Our analysis is thus complete, and can easily be extended to any finite number of coin tosses. The situation gets rapidly more complicated if the coin is tossed endlessly. This experiment cannot be conceived, carried out, or justified "in practice," and our neat conclusions would be rendered meaningless if we were unable to mathematically model our procedure. Happily, however, this is not the case. We

s and T<sup>0</sup> f gs

s and 0<sup>0</sup> f gs

instructive... to see the show in rehearsal as well as in performance."

*<sup>n</sup>*¼<sup>1</sup> of independent and identi-

appeared each time, the "null" statistical hypothesis

*Number Theory and Its Applications*

by all coins with *P H*ð Þ>0Þ, with a probability of 1.

**3. A probability model for infinite coin tossing**

teaching of probability; for example, sequences f g *Xn* <sup>∞</sup>

the probability *P A*ð Þ of *any* subset *A* may be defined by

tails.

sample space

simply let

**42**

for each *j* ¼ 0, 1, 2, *::::*9. Borel's law states that

*P X*ð is normalÞ ¼ 1,

which is somewhat surprising, since it is awfully hard to think of a single number that *is* normal (the number 0.012345678910111213..., obtained by writing each integer successively, is known to be normal; the proof is not trivial).

The Borel-Cantelli lemma yields several consequences that may, at first glance, seem to contradict Borel's normal number law:

Almost all the numbers in [0,1] (i.e., all except some with zero Lebesgue measure) have decimal expansions that contain infinitely many chains of length 1000, say, that contain no numbers except 2,3, and 4. The nice part is, of course, that almost all of these numbers are normal *as well*, and so on.

The moral of the Borel-Cantelli lemma should, by now, be quite clear: "The realization of a truly random infinite procedure will, with probability one, contain infinitely many segments that exhibit extreme 'non-randomness', of all sizes, patterns and intensities." The Borel-Cantelli lemma is, after all, a *limit theorem* of probability, and a quote from the classic treatise of Gnedenko and Kolmogorov [9] might be in order as well: "In reality, however, the epistemological value of the theory of probability is revealed only by limit theorems. Moreover, without limit theorems, it is impossible to understand the real content of the primary concept of all our sciences—the concept of probability."

#### **4. Conclusions and future developments**

The main results of this chapter, accessible to a second-year undergraduate, are Corollaries 2.3 and 2.5. They follow from the Borel-Cantelli lemmas and Boole's inequality, respectively. Corollary 2.3 states that in an infinite sequence of keystrokes, any fixed-length "work" appears infinitely often with probability 1. Most undergraduates that the author has taught have great difficulty believing this fact, since most statistical tests, for example, are based on finite samples. Corollary 2.5 goes one step further, proving that *every* finite-length piece of work, even those yet unwritten, will *each* appear infinitely often with probability 1. The undergraduate reader will undoubtedly appreciate the "power of infinity" on reading this chapter, while graduate students will enjoy a nonpractical yet deep application of the Borel-Cantelli lemmas.

Example 4 makes a contrast between the finite situation and the infinite one. An important practical problem in this regard would be to use Poisson approximations as in [10] to find the approximate probability that a specific work occurs *x* times in *n* keystrokes and to use this process as the basis of a statistical test for randomness.

**Author details**

Anant P. Godbole

**45**

East Tennessee State University, Johnson City, USA

\*Address all correspondence to: godbolea@etsu.edu

provided the original work is properly cited.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets…*

*DOI: http://dx.doi.org/10.5772/intechopen.93121*

*The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets… DOI: http://dx.doi.org/10.5772/intechopen.93121*

#### **Author details**

for each *j* ¼ 0, 1, 2, *::::*9. Borel's law states that

*Number Theory and Its Applications*

seem to contradict Borel's normal number law:

all our sciences—the concept of probability."

**4. Conclusions and future developments**

Cantelli lemmas.

**44**

*P X*ð is normalÞ ¼ 1,

The Borel-Cantelli lemma yields several consequences that may, at first glance,

which is somewhat surprising, since it is awfully hard to think of a single number that *is* normal (the number 0.012345678910111213..., obtained by writing

Almost all the numbers in [0,1] (i.e., all except some with zero Lebesgue measure) have decimal expansions that contain infinitely many chains of length 1000, say, that contain no numbers except 2,3, and 4. The nice part is, of course,

The moral of the Borel-Cantelli lemma should, by now, be quite clear: "The realization of a truly random infinite procedure will, with probability one, contain infinitely many segments that exhibit extreme 'non-randomness', of all sizes, patterns and intensities." The Borel-Cantelli lemma is, after all, a *limit theorem* of probability, and a quote from the classic treatise of Gnedenko and Kolmogorov [9] might be in order as well: "In reality, however, the epistemological value of the theory of probability is revealed only by limit theorems. Moreover, without limit theorems, it is impossible to understand the real content of the primary concept of

The main results of this chapter, accessible to a second-year undergraduate, are Corollaries 2.3 and 2.5. They follow from the Borel-Cantelli lemmas and Boole's inequality, respectively. Corollary 2.3 states that in an infinite sequence of keystrokes, any fixed-length "work" appears infinitely often with probability 1. Most undergraduates that the author has taught have great difficulty believing this fact, since most statistical tests, for example, are based on finite samples. Corollary 2.5 goes one step further, proving that *every* finite-length piece of work, even those yet unwritten, will *each* appear infinitely often with probability 1. The undergraduate reader will undoubtedly appreciate the "power of infinity" on reading this chapter, while graduate students will enjoy a nonpractical yet deep application of the Borel-

Example 4 makes a contrast between the finite situation and the infinite one. An important practical problem in this regard would be to use Poisson approximations as in [10] to find the approximate probability that a specific work occurs *x* times in *n* keystrokes and to use this process as the basis of a statistical test for randomness.

each integer successively, is known to be normal; the proof is not trivial).

that almost all of these numbers are normal *as well*, and so on.

Anant P. Godbole East Tennessee State University, Johnson City, USA

\*Address all correspondence to: godbolea@etsu.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Gamow G. One Two Three Infinity. New York: Bantam Books; 1971

**Chapter 4**

**Abstract**

2 ffiffiffiffi *N* <sup>p</sup> integers.

group is less than 2 ffiffiffiffi

alternations, fractal

**1. Introduction**

*N*

goes fractal = *Р1*# (see line 1 **Table 1**).

(primorial = *Р1*#) [1–6].

**47**

Prime Numbers Distribution Line

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on

are arranged in groups, by exactly three consecutive prime numbers of the form: (*Р*1-*Р*2-*Р*3). In this case, the distance from the first to the third prime number of any

prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than

**Keywords:** residue groups, prime numbers, primorial, sieve of Eratosthenes,

**1.1 Line-symmetrical primary-repeatable fractals of the positive integers**

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 pp. 109–116], positive integers are analyzed, hereinafter the P.I. is represented only as the alternance (array) of primes (according to the 1st least prime factor > 1 from every whole number). Type: 12.3.5.7.3.11.13.3.17.19.3.23.5.3.29 … 3.*р*.*р*.3.*р* … 3.*р*.*р*.3.*р* … . with for every recurrent prime = *Р*1, sieve of Eratosthenes formats the P.I., represented by alternance (array) of the first primes ≤*Р1*, in the form line-symmetrical repeating fractal-like structure, situated in the section of P.I. from 1 to *Р1*#, with "eliminated" sections of P.I. and φ(*Р1*#) not eliminated odd numbers are line-symmetrical to the number = *Р1*#/2 and are repeated without rearrangement of their position with the period = *Р1*#, on the basis of rhythmical repeating of two even numbers. Every recurrent prime has its own line-symmetrical primary-repeatable fractal *= Р1*, then

Every recurrent line-symmetrical fractal -*Р1*# is situated on the section of P.I. from 1 to *Р1*# and contains φ(*Р1*#) of the not eliminated odd numbers that are φ(*Р1*#) of the least residue, belonging to the indicated residue system (I.R.S) according to mod (*Р1*#), type: С<sup>n</sup> to the left from the number = *Р1*#/2 and **(***Р1*#**–**Сn) to the right from the number = *Р1*#/2, with С<sup>n</sup> – is residue according to mod(*Р1*#). Hereinafter with the term mod(*Р1*#), we shall indicate the period of fractal *Р1*# repetition (I.R.S, sieve of Eratosthenes), equal to product of all first primes ≤*Р<sup>1</sup>*

<sup>p</sup> integers, that is, *<sup>Р</sup>*3–*Р*<sup>1</sup> <sup>&</sup>lt; <sup>2</sup> ffiffiffiffiffi

*N*

<sup>p</sup> integers. (2) These same

*N*

<sup>p</sup> : (1) prime numbers

*Shcherbakov Aleksandr Gennadiyevich*

the segment of the natural series of numbers from 1 to N + 2 ffiffiffiffi

[2] Ash RB. Real Analysis and Probability. New York: Academic Press; 1972

[3] Borel E. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo. 1909;**27**:247-271

[4] Cantelli FP. Su due applicazioni di un teorema di G. Boole. Reale Academia Nazionale dei Lincei. 1917;**26**:295-302

[5] Balakrishnan N, Stepanov A. Generalization of the Borel-Cantelli lemma. The Mathematical Scientist. 2010;**35**:6162

[6] Stepanov A. On strong convergence. Communications in Statistics, Theory and Methods. 2015;**44**:1615-1620

[7] Petrov VV. A generalization of the Borel-Cantelli lemma. Statistics & Probability Letters. 2004;**67**:233-239

[8] Billingsley P. Probability and Measure. New York: John Wiley & Sons, Inc.; 1978

[9] Gnedenko BV, Kolmogorov AN. Limit Theorems for Sums of Independent Random Variables. Boston: Addison Wesley; 1954

[10] Godbole A, Schaffner A. Improved Poisson approximations for word patterns. Advances in Applied Probability. 1993;**25**:334-347

#### **Chapter 4**

**References**

1972

[1] Gamow G. One Two Three Infinity.

Probability. New York: Academic Press;

[4] Cantelli FP. Su due applicazioni di un teorema di G. Boole. Reale Academia Nazionale dei Lincei. 1917;**26**:295-302

[6] Stepanov A. On strong convergence. Communications in Statistics, Theory and Methods. 2015;**44**:1615-1620

[7] Petrov VV. A generalization of the Borel-Cantelli lemma. Statistics & Probability Letters. 2004;**67**:233-239

Measure. New York: John Wiley & Sons,

Independent Random Variables. Boston:

[10] Godbole A, Schaffner A. Improved Poisson approximations for word patterns. Advances in Applied Probability. 1993;**25**:334-347

[8] Billingsley P. Probability and

[9] Gnedenko BV, Kolmogorov AN. Limit Theorems for Sums of

Addison Wesley; 1954

[5] Balakrishnan N, Stepanov A. Generalization of the Borel-Cantelli lemma. The Mathematical Scientist.

2010;**35**:6162

Inc.; 1978

**46**

New York: Bantam Books; 1971

*Number Theory and Its Applications*

[2] Ash RB. Real Analysis and

[3] Borel E. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo. 1909;**27**:247-271

## Prime Numbers Distribution Line

*Shcherbakov Aleksandr Gennadiyevich*

#### **Abstract**

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2 ffiffiffiffi *N* <sup>p</sup> : (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (*Р*1-*Р*2-*Р*3). In this case, the distance from the first to the third prime number of any group is less than 2 ffiffiffiffi *N* <sup>p</sup> integers, that is, *<sup>Р</sup>*3–*Р*<sup>1</sup> <sup>&</sup>lt; <sup>2</sup> ffiffiffiffiffi *N* <sup>p</sup> integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2 ffiffiffiffi *N* <sup>p</sup> integers.

**Keywords:** residue groups, prime numbers, primorial, sieve of Eratosthenes, alternations, fractal

#### **1. Introduction**

#### **1.1 Line-symmetrical primary-repeatable fractals of the positive integers**

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 pp. 109–116], positive integers are analyzed, hereinafter the P.I. is represented only as the alternance (array) of primes (according to the 1st least prime factor > 1 from every whole number). Type: 12.3.5.7.3.11.13.3.17.19.3.23.5.3.29 … 3.*р*.*р*.3.*р* … 3.*р*.*р*.3.*р* … . with for every recurrent prime = *Р*1, sieve of Eratosthenes formats the P.I., represented by alternance (array) of the first primes ≤*Р1*, in the form line-symmetrical repeating fractal-like structure, situated in the section of P.I. from 1 to *Р1*#, with "eliminated" sections of P.I. and φ(*Р1*#) not eliminated odd numbers are line-symmetrical to the number = *Р1*#/2 and are repeated without rearrangement of their position with the period = *Р1*#, on the basis of rhythmical repeating of two even numbers. Every recurrent prime has its own line-symmetrical primary-repeatable fractal *= Р1*, then goes fractal = *Р1*# (see line 1 **Table 1**).

Every recurrent line-symmetrical fractal -*Р1*# is situated on the section of P.I. from 1 to *Р1*# and contains φ(*Р1*#) of the not eliminated odd numbers that are φ(*Р1*#) of the least residue, belonging to the indicated residue system (I.R.S) according to mod (*Р1*#), type: С<sup>n</sup> to the left from the number = *Р1*#/2 and **(***Р1*#**–**Сn) to the right from the number = *Р1*#/2, with С<sup>n</sup> – is residue according to mod(*Р1*#). Hereinafter with the term mod(*Р1*#), we shall indicate the period of fractal *Р1*# repetition (I.R.S, sieve of Eratosthenes), equal to product of all first primes ≤*Р<sup>1</sup>* (primorial = *Р1*#) [1–6].


Р2 *repeating of periodical fractal =* **Р1***#, including I.R.S. according to the mod(*Р1*#).*

By term residue according to mod (*Р1*#), we shall indicate every number, NOT

Fractal (Р2#)-I.R.S. mod(Р2#) = Р<sup>2</sup> lines in **Table 1**. -φ(Р1#) numbers multiple

Fractal (Р3#) I.R.S. mod(Р3#) = Р<sup>3</sup> lines in **Table 2**.-φ(Р2#) numbers multiple

Fractal (Р4#) I.R.S. mod(Р4#) = (Р<sup>4</sup> repeating of fractal Р3#)–φ(Р3#) numbers

Fractal (Р5#) I.R.S. mod(Р5#) = (Р<sup>5</sup> repeating of fractal Р4#)–φ(Р4#) numbers

**1.2 Purpose and role of the overall length of the of alternance (array) of the all**

It is quite obvious that φ(*Р*n#) of the least residues of mod(*Р*n#) type = С and **(***Р*n#**–**С), of every recurrent fractal = *Р*n#, gradate P.I. as φ(*Р*n#) "eliminated" sections of P.I. with different lengths of the type: *С*..3рр3.*С*.3рр3.*С* 3рр3 *С*., with ..3рр3.. "eliminated" sections of P.I. represented as array of "eliminated" NOT residues of mod(*Р*n#), or un the form of alternance (array) of the first primes ≤ *Р*n, (according to the 1st least prime factor >1 from every NOT residue of mod(*Р*n#)), hereinafter the alternance ≤*Р*n. С – residue of mod(*Р*n#) (according to the 1st least >*Р*<sup>n</sup> from every residue of mod(*Р*n#)), location from 1 to *Р*n# is line symmetrical relating to number = *Р*n#/2. And further, repeated without rearrangement of their position with the period = *Р*n#. Then, after we define the overall - maximal length of alternance that we can form using the fist primes р ≤ *Р***n**, (NOT residues of mod (*Р***n**#)), type С<sup>1</sup> … 3рр3рр3рр3 … С<sup>2</sup> that is maximal amount of consequent odd numbers = maximal length of alternance -р ≤ *Р***<sup>n</sup>** (one least NOT residue of mod (*Р***n**#) >1 from the number), we can evaluate the distance between every two consequent residues of mod (*Р***n**#) that is between two primes <(*Р***n+1**)

to formula: *(С<sup>2</sup> –С1)*–2/2 of the odd numbers ≤ maximal length of the alternance,

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 pp.109–116], the distribution of groups of 4 consequent residues in the form of "pairs of residues every two residue" is analyzed. But we have no information on distribution of groups of 4, 3,

In this scientific work, the φ(*Р*n#) of the least residue of mod(*Р*n#) of every recurrent fractal -*Р*n# is indexed as continuous sequence of groups: (а) No 4 has got 4 residues, or (b) No 3 has got 3 residues or (с) No 2 has got 2 consequent residues mod(*Р*n#). These groups No 4-3-2 are analyzed as subgroups with No 4-3-2 consequent residues of mod(*Р***n**#) that are surrounded by the maximal permissible

We used the mathematical induction method to define the overall – maximal length of every kind of subgroups No 4, No 3, No 2 and overall maximal length of P.

2

, according

Alternance ≤*Р<sup>1</sup>* is the section of P.I. in the form of array of primes – NOT residues of mod(*Р1*#), (for the 1 least common factor > 1 from every NOT residue). Eliminating (according to diagonals) 1 number multiple to Р<sup>2</sup> in every col-

eliminated by Sieve of Eratosthenes, not aliquot to the first primes ≤*Р1.*

umn = <sup>С</sup>n, we'll get in <sup>Р</sup><sup>2</sup> lines of **Table 1**: <sup>φ</sup>(Р1#)\*(Р<sup>2</sup> lines)**─**φ(Р1#)multiple Р<sup>2</sup> = φ(Р2#) residue of mod(Р2#). Representing the section of P.I. from 1 to Р2# as one line, we'll get the fractal = Р2# with the period of repeating =Р2#. And so on: every recurrent prime = Р<sup>n</sup> has got its periodical fractal = Рn# with n is the whole.

The numerical illustration is indicated in the scientific works [7–10].

Fractal (Р1#)-I.R.S. mod(Р1#) = (first line of **Table 1**).

and so on according to cumulative primes.

(maximal amount of NOT residue of mod(*Р***n**#)).

amount of consequent NOT residues of mod(*Р***n**#).

and 2 consequent residues of mod(*Р*n#) for every fractal *Р***n**#.

to Р2.

to Р3.

**49**

multiple to Р4.

multiple to Р5.

**first primes** ≤**Рn**

*Prime Numbers Distribution Line*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

*Prime Numbers Distribution Line DOI: http://dx.doi.org/10.5772/intechopen.92639*

By term residue according to mod (*Р1*#), we shall indicate every number, NOT eliminated by Sieve of Eratosthenes, not aliquot to the first primes ≤*Р1.*

Alternance ≤*Р<sup>1</sup>* is the section of P.I. in the form of array of primes – NOT residues of mod(*Р1*#), (for the 1 least common factor > 1 from every NOT residue).

Eliminating (according to diagonals) 1 number multiple to Р<sup>2</sup> in every column = <sup>С</sup>n, we'll get in <sup>Р</sup><sup>2</sup> lines of **Table 1**: <sup>φ</sup>(Р1#)\*(Р<sup>2</sup> lines)**─**φ(Р1#)multiple Р<sup>2</sup> = φ(Р2#) residue of mod(Р2#). Representing the section of P.I. from 1 to Р2# as one line, we'll get the fractal = Р2# with the period of repeating =Р2#. And so on: every recurrent prime = Р<sup>n</sup> has got its periodical fractal = Рn# with n is the whole. The numerical illustration is indicated in the scientific works [7–10].

Fractal (Р1#)-I.R.S. mod(Р1#) = (first line of **Table 1**).

Fractal (Р2#)-I.R.S. mod(Р2#) = Р<sup>2</sup> lines in **Table 1**. -φ(Р1#) numbers multiple to Р2.

Fractal (Р3#) I.R.S. mod(Р3#) = Р<sup>3</sup> lines in **Table 2**.-φ(Р2#) numbers multiple to Р3.

Fractal (Р4#) I.R.S. mod(Р4#) = (Р<sup>4</sup> repeating of fractal Р3#)–φ(Р3#) numbers multiple to Р4.

Fractal (Р5#) I.R.S. mod(Р5#) = (Р<sup>5</sup> repeating of fractal Р4#)–φ(Р4#) numbers multiple to Р5.

and so on according to cumulative primes.

#### **1.2 Purpose and role of the overall length of the of alternance (array) of the all first primes** ≤**Рn**

It is quite obvious that φ(*Р*n#) of the least residues of mod(*Р*n#) type = С and **(***Р*n#**–**С), of every recurrent fractal = *Р*n#, gradate P.I. as φ(*Р*n#) "eliminated" sections of P.I. with different lengths of the type: *С*..3рр3.*С*.3рр3.*С* 3рр3 *С*., with ..3рр3.. "eliminated" sections of P.I. represented as array of "eliminated" NOT residues of mod(*Р*n#), or un the form of alternance (array) of the first primes ≤ *Р*n, (according to the 1st least prime factor >1 from every NOT residue of mod(*Р*n#)), hereinafter the alternance ≤*Р*n. С – residue of mod(*Р*n#) (according to the 1st least >*Р*<sup>n</sup> from every residue of mod(*Р*n#)), location from 1 to *Р*n# is line symmetrical relating to number = *Р*n#/2. And further, repeated without rearrangement of their position with the period = *Р*n#. Then, after we define the overall - maximal length of alternance that we can form using the fist primes р ≤ *Р***n**, (NOT residues of mod (*Р***n**#)), type С<sup>1</sup> … 3рр3рр3рр3 … С<sup>2</sup> that is maximal amount of consequent odd numbers = maximal length of alternance -р ≤ *Р***<sup>n</sup>** (one least NOT residue of mod (*Р***n**#) >1 from the number), we can evaluate the distance between every two consequent residues of mod (*Р***n**#) that is between two primes <(*Р***n+1**) 2 , according to formula: *(С<sup>2</sup> –С1)*–2/2 of the odd numbers ≤ maximal length of the alternance, (maximal amount of NOT residue of mod(*Р***n**#)).

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 pp.109–116], the distribution of groups of 4 consequent residues in the form of "pairs of residues every two residue" is analyzed. But we have no information on distribution of groups of 4, 3, and 2 consequent residues of mod(*Р*n#) for every fractal *Р***n**#.

In this scientific work, the φ(*Р*n#) of the least residue of mod(*Р*n#) of every recurrent fractal -*Р*n# is indexed as continuous sequence of groups: (а) No 4 has got 4 residues, or (b) No 3 has got 3 residues or (с) No 2 has got 2 consequent residues mod(*Р*n#). These groups No 4-3-2 are analyzed as subgroups with No 4-3-2 consequent residues of mod(*Р***n**#) that are surrounded by the maximal permissible amount of consequent NOT residues of mod(*Р***n**#).

We used the mathematical induction method to define the overall – maximal length of every kind of subgroups No 4, No 3, No 2 and overall maximal length of P.

С1 = 1

**48**

**1 +**

*Р1***#** **1+2***Р1***#**

**…** **…** **1 +** And so on, repeating of fractal =

**Table 1.**

Р2

*repeating of periodical*

 *fractal =* **Р1***#,*

*including I.R.S. according to the mod(*Р1*#).*

*Р2***#**

3,5,7..

С2 + *Р2*# *Р1*# with the period =

*Р1*#, with: ррр is alternance of ≤ *Р*1

 pрр

С3 + *Р2*#

ррр

Сn +

*Р2*#

ррр

3,5,7..

…

pрр

...

ррр

…

ррр

*Р2*#–Сn

ррр

ррр

*Р2*#–С2

**..7,5.3** **..7,5.3**

**(***Р2***#–1)**

3,5,7..

…

pрр

…

ррр

…

ррр

*…*

ррр

…

**..7,5.3**

…

3,5,7..

С2 + 2*Р1*#

 pрр

С3 + 2*Р1*#

ррр

Сn+2*Р1*#

ррр

*3Р1*#–Сn

ррр

3*Р1*#–С2

**..7,5.3**

**(3***Р1***#–1)**

3,5,7..

С2 + *Р1*#

 pрр

С3 + *Р1*#

ррр

Сn +

*Р1*#

ррр

2*Р1*#–Сn

ррр

2*Р1*#–С2

**..7,5.3**

**(2***Р1***#–1)**

 3,5,7..

С2 = *Р2*

**pрр**

..**С3** ..

**ррр**

.. **Сn** ..

**ррр**

*Р1***#–Сn**

**ррр**

*Р1***#–С2**

..7,5.3

(*Р1*#–1)

*Number Theory and Its Applications*


**Table**

**2.**

 Р<sup>3</sup>*repeating of periodical fractal =* **Р2***#, including I.R.S. according to the mod(*Р2*#).*

I. sections in the form of maximal long alternances of all first primes ≤Рn, (that is maximal permissible amount of all NOT residues of mod(*Р***n**#)), situated between two residues from СА to СВ, between which, as subgroups are situated the groups of

As a result, we detected the loopback of these groups rearrangement from No 4 to No 3 up to No 2 according to the growing amount of the modulus, and the primes

**2. Three groups of "eliminated"sections of every next fractal**

different first primes ≤*Р1*, NOT residues of mod(*Р1*#), type:

It is quite obvious and requires no proof that indexing φ(*Р1*#) of the least residues of mod(*Р1*#) of every recurrent fractal-*Р1*#, or I.R.S mod(*Р1*#), is by groups, containing strictly 4 elements; three; two consequent residues of mod(*Р1*#), we have, that every recurrent fractal-*Р1*# would be represented as array of three groups of the residues of mod(*Р1*#), between are situated the alternances ≤*Р<sup>1</sup>* (with different lengths) – the consequent NOT residues of mod(*Р1*#), **of types** (**а), (b),**

a. φ(*Р1*#) groups No 4 containing strictly FOUR consequent residues of mod (*Р1*#)A,B,C,D **С***,* between which the alternances of different amounts of

*<sup>A</sup>С*..3рр3..*BС..*3рр3..*CС* .. 3рр3..D*С*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod(*Р1*#). (1–4С) – Four consequent residue of mod(*Р*1#), including the consequent primes of P.I. section from *Р*<sup>1</sup> to (*Р2)2* of A,B,C,D*Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 4 groups (4 residues) of mod (*Р1*#). Three adjoined groups No 4 for every residue = С (**Table 3**).

The length of group No 4, which means amount odd numbers, restricted by every group No 4 from <sup>A</sup>*P* to <sup>D</sup>*P* and from С<sup>1</sup> to С4, for the mod(*Р1*#), is (R4– 2)/2 ≤ (*Р*2–1) of the odd numbers with: R4 = (D*P–*A*P*), R4 = (4С–1С), R4 ≤ 2*Р*<sup>2</sup> (including 1 group of R4 = 2*Р*2, detailed information is indicated in Section 5)

b. φ(*Р1*#) groups No 3 containing strictly THREE consequent residues of mod (*Р1*#). A,B,C**С***,* between which the alternances of different amounts of

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 4 groups (containing 4 residues) of mod(*Р1*#). Three adjoined*

*<sup>2</sup>* <sup>1</sup>С.3рр3.4С.3рр3.. *Р1*#

*<sup>2</sup>* —2С..3рр3..5С..2C *Р1*#

*<sup>2</sup>* ———3С..3рр3..6С..3C *Р1*#

residues of mod (*Р*n#). Type:

*Prime Numbers Distribution Line*

order distribution is defined.

(**Table 4**).

*groups No 4 for every residue = С.*

**Table 3.**

**51**

a. -No 4: СА..3рр3..*Р1..Р2..Р3..Р4*..3рр3..СВ.

**(с)** repeated without changes with period = *Р1*#.

1,3,5,7 … <sup>A</sup>*P* ..3рр3..B*P*..3рр3..C*P*..3рр3..D*P*. 3рр3... *P2*

3,5,7 … *—————*B*P*..3рр3..C*P*..3рр3..D*P*..3рр3..A*P* … *P2*

5,7 … *- ———*C*P.*.3рр3..D*P*.3рр3..A*P*..3рр3..B*P P2*

*And so on, repeating of fractal = Р1# with the period = Р1#, ррр - is alternance of* ≤ *Р1.*

b. No 3: СА..3рр3..*Р1..Р2..Р*3..3рр3..СВ.

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

c. No 2: СА..3 рр3..*Р1..Р2..3*рр3..*С*В.

I. sections in the form of maximal long alternances of all first primes ≤Рn, (that is maximal permissible amount of all NOT residues of mod(*Р***n**#)), situated between two residues from СА to СВ, between which, as subgroups are situated the groups of residues of mod (*Р*n#). Type:


As a result, we detected the loopback of these groups rearrangement from No 4 to No 3 up to No 2 according to the growing amount of the modulus, and the primes order distribution is defined.

#### **2. Three groups of "eliminated"sections of every next fractal**

It is quite obvious and requires no proof that indexing φ(*Р1*#) of the least residues of mod(*Р1*#) of every recurrent fractal-*Р1*#, or I.R.S mod(*Р1*#), is by groups, containing strictly 4 elements; three; two consequent residues of mod(*Р1*#), we have, that every recurrent fractal-*Р1*# would be represented as array of three groups of the residues of mod(*Р1*#), between are situated the alternances ≤*Р<sup>1</sup>* (with different lengths) – the consequent NOT residues of mod(*Р1*#), **of types** (**а), (b), (с)** repeated without changes with period = *Р1*#.

a. φ(*Р1*#) groups No 4 containing strictly FOUR consequent residues of mod (*Р1*#)A,B,C,D **С***,* between which the alternances of different amounts of different first primes ≤*Р1*, NOT residues of mod(*Р1*#), type: *<sup>A</sup>С*..3рр3..*BС..*3рр3..*CС* .. 3рр3..D*С*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod(*Р1*#). (1–4С) – Four consequent residue of mod(*Р*1#), including the consequent primes of P.I. section from *Р*<sup>1</sup> to (*Р2)2* of A,B,C,D*Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 4 groups (4 residues) of mod (*Р1*#). Three adjoined groups No 4 for every residue = С (**Table 3**).

The length of group No 4, which means amount odd numbers, restricted by every group No 4 from <sup>A</sup>*P* to <sup>D</sup>*P* and from С<sup>1</sup> to С4, for the mod(*Р1*#), is (R4– 2)/2 ≤ (*Р*2–1) of the odd numbers with: R4 = (D*P–*A*P*), R4 = (4С–1С), R4 ≤ 2*Р*<sup>2</sup> (including 1 group of R4 = 2*Р*2, detailed information is indicated in Section 5) (**Table 4**).

b. φ(*Р1*#) groups No 3 containing strictly THREE consequent residues of mod (*Р1*#). A,B,C**С***,* between which the alternances of different amounts of


**Table 3.**

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 4 groups (containing 4 residues) of mod(*Р1*#). Three adjoined groups No 4 for every residue = С.*

С1 = 1

**50**

**1 +**

*Р2***#**

**1+2***Р2***#**

**…** **…** **1 +** *And so on, repeating of fractal =*

*Representing*

**Table 2.**

Р<sup>3</sup>

*repeating of periodical*

 *fractal =* **Р2***#,*

*including I.R.S. according to the mod(*Р2*#).*

 *the section of P.I. as line from 1 to*

*Р3***#**

 **3,5,7,,,**

С2 + *Р3*# *Р2# with the period =*

*Р2#, with: ррр is alternance of* ≤ *Р2.*

*Р3# we'll get the fractal*

*Р3# and so on.*

 pрр

С3 + *Р3*#

ррр

Сn +

*Р3*#

ррр

*…*

ррр

*…*

**..7,5.3**

…

**3,5,7,,,**

…

pрр

...

ррр

…

ррр

*Р3*#–Сn

ррр

*Р3*#–С2

**..7,5.3**

**(***Р3***#–1)**

**3,5,7,,,**

…

pрр

…

ррр

…

ррр

*…*

ррр

…

**..7,5.3**

…

 **3,5,7,,,**

С2 + 2*Р2*#

 pрр

С3 + 2*Р2*#

ррр

Сn+2*Р2*#

ррр

*3Р2*#–Сn

ррр

3*Р2*#–С2

**..7,5.3**

**(3***Р2***#–1)**

 **3,5,7,,,**

С2 + *Р2*#

 pрр

С3 + *Р2*#

ррр

Сn +

*Р2*#

ррр

2*Р2*#–Сn

ррр

2*Р2*#–С2

**..7,5.3**

**(2***Р2***#–1)**

 3,5,7,,,

С2 = *Р3*

**pрр**

..**С3** ..

**ррр**

.. **Сn** ..

**ррр**

*Р2***#–Сn**

**ррр**

*Р2***#–С2**

..7,5.3

(*Р2*#–1)

*Number Theory and Its Applications*


**Table 4.**

*The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 4 (containing four residues). The three adjoined groups No 4 for every residue = С with R4* ≤ *2*Р*<sup>2</sup> = 2\*7 (consult Section 5).*

different first prime ≤*Р1*, NOT residues of mod (*Р1*#), type: *<sup>A</sup>С*..3рр3..*BС..*3рр3..*CС*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod (*Р1*#). (1-3С) – three consequent residues of mod(*Р*1#), including the consequent primes of P.I. section from *Р*<sup>1</sup> to (*Р2)2* of A,B,C *Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 3 groups (3 residues) of mod(*Р1*#). Two adjoined groups No 3 for every residue = С (**Table 5**).

With the unknown to us, length of the group No 2 from <sup>A</sup>*P* to <sup>B</sup>*P* and from С<sup>1</sup> to

*<sup>2</sup>* **<sup>1</sup>С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С...** *Р2***#**

*<sup>2</sup>* **<sup>1</sup>С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С...** *Р3***#**

Herewith for each group No 4-3-2 according to mod(*Р1*#), there are two residues of mod(*Р1*#): СА – to the left and СВ – to the right, that is every group No 4-3-2 is the subgroup on the P.I. sections of the length unknown to us from СА to СВ: (а)СА-

С2, for the mod(*Р1*#), is (R2–2)/2 of the odd numbers with: R2 = (B*P–*A*P*)., R2 = (2С–1С), R2 =? (it is quite obvious that for mod(*Р1*#) R2 *< <*R3).

*And so on, repeating of fractal = Р2# with the period = Р2#, ррр - is alternance of* ≤ *Р2.*

*And so on, repeating of fractal = Р3# with the period = Р3#, ррр - is alternance of* ≤*Р3.*

**3. Correlations of length limits of the subgroups No 4, No 3, No 2**

(containing 4 residual for every recurrent fractal -*Р*n#), of type is defined:

**Max length of φ(***Р*n**#) of the subgroups No 4 maxR4 = (С4–С1)**

(С4–С1) = *2Р<sup>2</sup>*

(С4–С1) = *2Р<sup>3</sup>*

(С4–С1) = *2Р<sup>5</sup>*

(С4–С1) = *2Р<sup>6</sup>*

(С4–С1)=2*Р*n+1

*The relation of length limits of the subgroups according to the increasing modulus.*

And so on, repeating of fractal = *Р*n# with the period = *Р*n#. C1–<sup>4</sup> - residue of mod (*Р*n#).

**…** *…* **… …** >> **…** >> **…**

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 p. 109–116, 10 p. 1805], including Section 5 of this work, the overall – maximal length of the subgroup No 4

Herewith, it is quite obvious and it is beyond argument that relations of limits, unknown to us of groups No 4-3-2 length according to the increasing modulus are

> **>> Max length φ(***Р*n**#) of the subgroups No 3 maxR3 = (С3–С1)**

>> max R3 = (С3–С1) =?

>> max R3 = (С3–С1) =?

>> max R3 = (С3–С1) =?

>> max R3 = (С3–С1) =?

>> max R3 = (С3–С1) =? **>> Max length φ(***Р*n**#) of the subgroups No 2 maxR2 = (С2–С1)**

>> max R2 = (С2–С1) =?

>> max R2 = (С2–С1) =?

>> max R2 = (С2–С1) =?

>> max R2 = (С2–С1) =?

>> max R2= (С2–С1) =?

(С1-С2-С3-С4)-СВ. (b) СА-(С1-С2-С3)-СВ. (с) СА-(С1-С2)-СВ.

**1.,3.,5.,7..** *<sup>A</sup>Р***..3рр3..***BP***..3рр3..***АP***.3рр3..** *P3*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

*Prime Numbers Distribution Line*

**1.,3.,5.,7..** *<sup>A</sup>Р***..3рр3..***BP***..3рр3..***АP***.3рр3..** *P4*

*Fractal =* Р2*#, represented as φ(*Р2*#) of No 2 groups of mod(*Р2*#).*

*Fractal =* Р3*#, represented as φ(*Р3*#) of No 2 groups of mod(*Р3*#).*

**Table 8.**

**Table 9.**

max R4 = (С<sup>4</sup> – С1)=2*Р*n+1 of whole numbers.

**Period of fractal repetition =mod (***Р***n#).**

*Р*<sup>1</sup> *Р1*# mod(*Р1*#). max R4 =

*Р*<sup>2</sup> *Р2*# mod(*Р2*#). max R4 =

*Р*<sup>4</sup> *Р4*# mod(*Р4*#). max R4 =

*Р*<sup>5</sup> *Р5*# mod(*Р5*#). max R4 =

*Р*<sup>n</sup> *Р*n# mod(*Р*n#). max R4 =

indicated in **Table 10**.

**Fractal -** *Р***n#**

**Prime value** *Р***n**

**Table 10.**

**53**

With the unknown to us length of the group No 3 from <sup>A</sup>*P* to <sup>C</sup>*P* and from С<sup>1</sup> to С3, for the mod (*Р1*#), is (R3–2)/2 of the odd numbers with: R3 = (C*P–*A*P*)., R3 = (3С–1С)., R3 =? (it is quite obvious that for mod (*Р1*#) R3 *< <*R4) (**Table 6**).

c. φ(*Р1*#) groups No 2 containing strictly TWO consequent residues of mod (*Р1*#)A,B**С***,* between which the alternances of different amounts of different first prime ≤*Р1*, NOT residues of mod(*Р1*#), type: *<sup>A</sup>С*..3рр3..*BС*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod(*Р1*#). (1–2С) – two consequent residue of mod(*Р*1#), including the consequent primes of P.I. section from *Р*<sup>1</sup> to *(Р2)2* of A,B*Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 2 groups (2 residues) of mod(*Р1*#) (**Tables 7**–**9**).


**Table 5.**

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 3 groups (containing 3 residues) of mod(*Р1*#). Two adjoined groups No 3 for every residue = С.*


#### **Table 6.**

*The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 3 (containing two residues). The two adjoined groups No 3 for every residue = С with R3 =? (=2\*5 consult Section 6).*


#### **Table 7.**

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 2 groups of mod(*Р1*#).*


**Table 8.**

*Fractal =* Р2*#, represented as φ(*Р2*#) of No 2 groups of mod(*Р2*#).*

**1.,3.,5.,7..** *<sup>A</sup>Р***..3рр3..***BP***..3рр3..***АP***.3рр3..** *P4 <sup>2</sup>* **<sup>1</sup>С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С...** *Р3***#** *And so on, repeating of fractal = Р3# with the period = Р3#, ррр - is alternance of* ≤*Р3.*

**Table 9.**

different first prime ≤*Р1*, NOT residues of mod (*Р1*#), type:

*The three adjoined groups No 4 for every residue = С with R4* ≤ *2*Р*<sup>2</sup> = 2\*7 (consult Section 5).*

Type-(c) **С<sup>3</sup> = 11** 13 17 **С<sup>6</sup> = 19** 23 29 С = 31

*And so on, repeating of fractal =5# with the period = mod(5#)*

*Number Theory and Its Applications*

**Table 4.**

**Table 5.**

**Table 6.**

**Table 7.**

**52**

*groups No 3 for every residue = С.*

Type-(a) **С<sup>1</sup> = 1** 7 11 **С<sup>4</sup> = 13** 17 19 **С<sup>7</sup> = 23** 29 31 С = 37

Type-(b) **С<sup>2</sup> = 7** 11 13 **С<sup>5</sup> = 17** 19 23 **С<sup>8</sup> = 29** 31 37 С = 41

*The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 4 (containing four residues).*

Two adjoined groups No 3 for every residue = С (**Table 5**).

groups (2 residues) of mod(*Р1*#) (**Tables 7**–**9**).

*And so on, repeating of fractal = Р1# with the period = Р1#, with: ррр is alternance of* ≤ *Р1.*

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 3 groups (containing 3 residues) of mod(*Р1*#). Two adjoined*

Type-(b) **С<sup>2</sup> = 7** 11 **С<sup>4</sup> = 13** 17 **С<sup>6</sup> = 19** 23 **С<sup>8</sup> = 29** 31 С = 37

*The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 3 (containing two residues).*

Type-(a) **С<sup>1</sup> = 1** 7 **С<sup>3</sup> =** 11 13 **С<sup>5</sup> = 17** 19 **С<sup>7</sup> = 23** 29 С =31

*The two adjoined groups No 3 for every residue = С with R3 =? (=2\*5 consult Section 6).*

*And so on, repeating of fractal = Р1# with the period = Р1#, ррр - is alternance of*≤ *Р1.*

**1.,3.,5.,7..** *<sup>A</sup>Р***..3рр3..***BP* **..3рр3..***CP P2*

3.,5.,7 … *——————*.*BP* ..3рр3..*CP*..3рр3..А*P*..3рр3.. *P2*

*And so on, repeating of fractal =5# with the period =5#.*

**1.,3.,5.,7..** *<sup>A</sup>Р***..3рр3..***BP***..3рр3..***АP***.3рр3..** *P2*

*Fractal =* Р1*#, represented as φ(*Р1*#) of No 2 groups of mod(*Р1*#).*

*<sup>A</sup>С*..3рр3..*BС..*3рр3..*CС*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod

consequent primes of P.I. section from *Р*<sup>1</sup> to (*Р2)2* of A,B,C *Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 3 groups (3 residues) of mod(*Р1*#).

With the unknown to us length of the group No 3 from <sup>A</sup>*P* to <sup>C</sup>*P* and from С<sup>1</sup> to С3, for the mod (*Р1*#), is (R3–2)/2 of the odd numbers with: R3 = (C*P–*A*P*)., R3 = (3С–1С)., R3 =? (it is quite obvious that for mod (*Р1*#) R3 *< <*R4) (**Table 6**).

*<sup>2</sup>* **<sup>1</sup>С..3рр3..3С … 3рр3.. <sup>1</sup>С … <sup>3</sup>С** *Р1***#**

*<sup>2</sup>* ——————2С..3рр3..4С.. <sup>2</sup>С .. <sup>4</sup>С *Р1*#

*<sup>2</sup>* **<sup>1</sup>С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С...** *Р1***#**

(*Р1*#). (1-3С) – three consequent residues of mod(*Р*1#), including the

c. φ(*Р1*#) groups No 2 containing strictly TWO consequent residues of mod (*Р1*#)A,B**С***,* between which the alternances of different amounts of different first prime ≤*Р1*, NOT residues of mod(*Р1*#), type: *<sup>A</sup>С*..3рр3..*BС*. with: ..3рр3.. are alternances of the first primes ≤*Р*<sup>1</sup> according to the 1st least common factor > 1 from every NOT residue of mod(*Р1*#). (1–2С) – two consequent residue of mod(*Р*1#), including the consequent primes of P.I. section from *Р*<sup>1</sup> to *(Р2)2* of A,B*Р* type*.* Further, the fractal = *Р1*# represented as φ(*Р1*#) of No 2

*Fractal =* Р3*#, represented as φ(*Р3*#) of No 2 groups of mod(*Р3*#).*

With the unknown to us, length of the group No 2 from <sup>A</sup>*P* to <sup>B</sup>*P* and from С<sup>1</sup> to С2, for the mod(*Р1*#), is (R2–2)/2 of the odd numbers with: R2 = (B*P–*A*P*)., R2 = (2С–1С), R2 =? (it is quite obvious that for mod(*Р1*#) R2 *< <*R3).

Herewith for each group No 4-3-2 according to mod(*Р1*#), there are two residues of mod(*Р1*#): СА – to the left and СВ – to the right, that is every group No 4-3-2 is the subgroup on the P.I. sections of the length unknown to us from СА to СВ: (а)СА- (С1-С2-С3-С4)-СВ. (b) СА-(С1-С2-С3)-СВ. (с) СА-(С1-С2)-СВ.

#### **3. Correlations of length limits of the subgroups No 4, No 3, No 2**

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 p. 109–116, 10 p. 1805], including Section 5 of this work, the overall – maximal length of the subgroup No 4 (containing 4 residual for every recurrent fractal -*Р*n#), of type is defined:

max R4 = (С<sup>4</sup> – С1)=2*Р*n+1 of whole numbers.

Herewith, it is quite obvious and it is beyond argument that relations of limits, unknown to us of groups No 4-3-2 length according to the increasing modulus are indicated in **Table 10**.


#### **Table 10.**

*The relation of length limits of the subgroups according to the increasing modulus.*

#### **4. Distribution of prime number**

Correlation of length limits of the subgroups in **Table 10** and distribution of groups of the indexed residues No 4-3-2, in every respective fractal -*Р*n# according to the increasing modulus is defined by theorem 1.

**Theorem 1.** The loopback of prime number distribution.

Every prescribed prime number squared = *(Р2)<sup>2</sup>* defines the distribution of all previous prime numbers *< (Р2)2* , as all first prime numbers are less than every prescribed prime number squared = (*P2*) *2* , are situated in the P.I. as part of fractal *Р*1#, where they are distributed by subgroups of (**а), (b), (с)** types.

a. φ(*Р1*#) of subgroups No 4 having pure FOUR consequent prime number (A,B, C,DС) of *P1 <* (AС-BС*-*CС-DС) < *P2 2* ; *Р<sup>3</sup> <sup>2</sup>* type. At the P.I. section from *Р<sup>1</sup>* to *Р2 <sup>2</sup>* (including from *(Р2–2)<sup>2</sup>* to *Р<sup>2</sup> 2* ), and further, from *Р<sup>2</sup> <sup>2</sup>* to *Р1*# pure FOUR consequent residues of mod(*Р1*#) on all P.I. sections with length not exceeding 2*Р<sup>3</sup>* of whole numbers with length of every subgroup No 4 at every section is:

$$\mathbf{R\_4 = (\_DC\_{-AC}C)} \le 2\mathbf{P\_2}.$$

In that case, these primes of fractal = *Р1***#**, by loopback, are distributed by groups:


With: A,B,C,DС are consequent residues of mod(*Р1*#) including the primes < (*Р2)2* . R4-3-2 is the remainder of the first and the last number of every group No 4-3-2 (of the fractal *Р*1#). Further, the length of the subgroup No 4-3-2 as the amount of odd numbers, restricted by every group from AC to B-C-DC, from <sup>A</sup>*P* to B-C-D*P* are (R4,3,2–2)/2 odd numbers.

The order of groups (а), (b), (с) rearrangement according to the increasing modulus for visual clarity is indicated in **Table 11**.

#### **5. Proof of theorem**

#### **5.1 Proof of section a of the Theorem 1**

It is feasible that in P.I. using the first prime number ≤ *Р<sup>n</sup>* (NOT residues of mod (*Рn*#)), by the only single way, we can form the maximal long P.I. section as the maximal long alternance – array for the 1 least common factor > 1 from every NOT residue of mod(*Р***n**#). That is that maximal amount of NOT residues of mod(*Рn*#), maximal long alternance ≤*Р***n**.

**1**

**55**

Fractal from 1 to

Composition

repeating =mod(

*Рn*#)

of primes number of

this fractal including

on the P.I section:

Subgroup No 4

Length of every section

Subgroup No 3

Length of every section

Subgroup No 2

Length of every section

*Prime Numbers Distribution Line*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

for the group No 2

length. *R2 = BС–AС*

for the group No 3

length. *R3 = CС–AС*

for the group No 4

length. *R4 = DС–AС*

(*Р*n+1)

4*Р*n+1 of whole

numbers

1 ÷

…

1 ÷ 1 ÷ 1 ÷ 1 ÷ 1 ÷

*… and so on according to the increasing meanings of the modulus*

**Table 11.** *Loopback of primes number subgroups distribution*

 *according to the increasing meanings of the modulus.*

*Р5*# mod(

*Р5*#)

 from (

to (

*Р6*) *2*

number

*…*

*Р5*)

≥4*Р6*

*R4* ≤ 2*P6*

≤2*P7* number

*R3* ≤ 2*P5*

≤2*P6* number

*R2* ≤ 2*P4*

≤2*P5* number

*2*

*Р4*# mod(

*Р4*#)

 from (

to (

*Р5*) *2*

number

*Р4*)*2*

≥4*Р5*

*R4* ≤ 2*P5*

≤2P*6* number

*R3* ≤ 2*P4*

≤2*P5* number

*R2* ≤ 2*P3*

≤2*P4* number

*Р3*# mod(

*Р3*#)

 from (

to (

*Р4*) *2*

number

*Р3*)

≥4*Р4*

*R4* ≤ 2*P4*

≤2*P5* number

*R3* ≤ 2*P3*

≤2*P4* number

*R2* ≤ 2*P2*

≤2*P3* number

*2*

*Р2*# mod(

*Р2*#)

 from (

to (

*Р3*)

number

*2*

*Р2*)

≥4*Р3*

*R4* ≤ 2*P3*

≤2*P4* number

*R3* ≤ 2*P2*

≤2*P3* number

*R2* ≤ 2*P1*

≤2*P2* number

*2*

*Р1*# mod(

*Р1*#)

 from (

to (

*Р2*)

number

*2*

*Р1*)

≥4*Р2*

*R4* ≤ 2*P2*

≤2*P3* number

*2*

*Рn*# mod(

*Рn*#)

 from (

to (

*Рn+1*)

…

 …

*…*

…

*…*

*R3* ≤ 2*P1*

≤2*P2* number

*R2* ≤ 2*P0*

≤2*P1* number

*2*

number

*<*(*P*n# *Pn+1*)

*Рn*)*2*

≥4*Рn+1*

*R4* ≤ 2*Pn+1*

≤2*Pn+2 <*

*R3* ≤ 2*P*n

≤2*P*n+1 number

*R2* ≤ 2*Pn–1*

≤2*Pn* number

> (**Table 13**)

………

(**Table 12**)

(*P*n# *Pn+2*)

*2*

–(*Р*n+1*–*2)2 =

 and its

*Рn*#

Length of P.I. section

which defines values

**23**

φ(*Р*n#) groups No 4 for 4 residues of mod

(*Рn*#) (containing

 4 simple)

< *Pn+12*

AС-BС*-*CС-DС

(*Рn*#) (containing

 3 simple)

< *Pn+12*

AС-BС*-*CС

(*Рn*#) (containing

 2 simple) (AС-BС)

< *Pn+12*

φ(*Рn*#) groups No 2 for 2 residues of mod

φ(*Рn*#) groups No 3 for 3 residues of mod

 **4**

 **5**


**Table 11.** *Loopback of primes number subgroups distribution*

 *according to the increasing meanings of the modulus.*

#### *Prime Numbers Distribution Line DOI: http://dx.doi.org/10.5772/intechopen.92639*

**4. Distribution of prime number**

*Number Theory and Its Applications*

previous prime numbers *< (Р2)2*

*Р2*

section is:

groups:

further, from *Р<sup>2</sup>*

further, from *Р<sup>2</sup>*

(R4,3,2–2)/2 odd numbers.

**5. Proof of theorem**

maximal long alternance ≤*Р***n**.

**54**

prescribed prime number squared = (*P2*)

C,DС) of *P1 <* (AС-BС*-*CС-DС) < *P2*

*<sup>2</sup>* (including from *(Р2–2)<sup>2</sup>* to *Р<sup>2</sup>*

(A,B,CС) of *P1 <* (AС-BС*-*CС) < *P2*

(A,BС) of *P0*, *P1 <* (AС-BС) < *P2*

modulus for visual clarity is indicated in **Table 11**.

**5.1 Proof of section a of the Theorem 1**

to the increasing modulus is defined by theorem 1.

**Theorem 1.** The loopback of prime number distribution.

*Р*1#, where they are distributed by subgroups of (**а), (b), (с)** types.

Correlation of length limits of the subgroups in **Table 10** and distribution of groups of the indexed residues No 4-3-2, in every respective fractal -*Р*n# according

Every prescribed prime number squared = *(Р2)<sup>2</sup>* defines the distribution of all

a. φ(*Р1*#) of subgroups No 4 having pure FOUR consequent prime number (A,B,

exceeding 2*Р<sup>3</sup>* of whole numbers with length of every subgroup No 4 at every

*2* ; *Р<sup>3</sup>*

R4 ¼ ðDС–AСÞ≤2Р2*:*

In that case, these primes of fractal = *Р1***#**, by loopback, are distributed by

b. φ(*Р1*#) of subgroups No 3 having pure THREE consequent prime number

c. φ(*Р1*#) of subgroups No 2 having pure TWO consequent prime number

I. sections with length not exceeding 2*Р<sup>1</sup>* of whole numbers with length of

With: A,B,C,DС are consequent residues of mod(*Р1*#) including the primes < (*Р2)2*

R4-3-2 is the remainder of the first and the last number of every group No 4-3-2 (of the fractal *Р*1#). Further, the length of the subgroup No 4-3-2 as the amount of odd numbers, restricted by every group from AC to B-C-DC, from <sup>A</sup>*P* to B-C-D*P* are

The order of groups (а), (b), (с) rearrangement according to the increasing

It is feasible that in P.I. using the first prime number ≤ *Р<sup>n</sup>* (NOT residues of mod (*Рn*#)), by the only single way, we can form the maximal long P.I. section as the maximal long alternance – array for the 1 least common factor > 1 from every NOT residue of mod(*Р***n**#). That is that maximal amount of NOT residues of mod(*Рn*#),

every subgroup No 3 at every section is: R3 = (CС–AС) ≤ **2***Р1*.

every subgroup No 2 at every section is: R2 = (BС–AС) ≤ **2***Р0*.

P.I. sections with length not exceeding 2*Р<sup>2</sup>* of whole numbers with length of

*2*

consequent residues of mod(*Р1*#) on all P.I. sections with length not

*2*

, as all first prime numbers are less than every

), and further, from *Р<sup>2</sup>*

, are situated in the P.I. as part of fractal

*<sup>2</sup>* type. At the P.I. section from *Р<sup>1</sup>* to

*<sup>2</sup>* type. At the P.I. section from *Р<sup>1</sup>* to *Р<sup>2</sup>*

*<sup>2</sup>* type. At the P.I. section from *Р<sup>1</sup>* to *Р<sup>2</sup>*

*<sup>2</sup>* to *Р1*# pure THREE consequent residues of mod(*Р1*#) on all

*<sup>2</sup>* to *Р1*# pure TWO consequent residues of mod(*Р1*#) on all P.

*<sup>2</sup>* to *Р1*# pure FOUR

*<sup>2</sup>* and

*<sup>2</sup>* and

.

**55**

Then for every recurrent prime number *Р<sup>n</sup>* = *Р<sup>1</sup>* at the P.I., formed as recurrent line symmetrical, primary-repeatable periodical fractal = Р1# or I.R.S. according to mod(Р1#), (see the first line of **Table 1**), the maximal long P.I. section, formed as the alternance of the all first primes number ≤ *Р1*, (NOT residues of mod(*Рn*#)), shall be situated within the P.I. section from СА to СВ, with as the subgroup is the only maximal long maximal subgroup No 4 (С1-С2-С3-С4) with the 4 consequent residue of mod(*Р1*#): Type: СА = (*Р1*#–*Р3*)..С<sup>1</sup> = (*Р1*#–*Р2*).

subgroup No 4 (containing 4 residues) of mod(*Р(1)*#) with length = R4 > 2\**Р(2)*, situated within the alternance of all first primes number ≤ *Р(1)* within the P.I. section with length > 2\**Р(3),* >(*Р(3)*–1) of the odd numbers. Then every subgroup No 4 would be line-symmetrical to the left and to the right from the center of the fractal *Р(1)*# symmetry of the number *Р(1)*#/2. That is, in the result, we'll get in fractal *Р(1)*# using all primes number ≤ *Р(1)*, we can by more than by one way from the maximally long alternance of all the prime numbers ≤ *Р(1)*, that is by the sieve of Eratosthenes, focused to the left and to the right (to the left and to the right

**6. The maximal length of P.I. section with maximal long subgroups No 3**

At the fractal -*Р1*#, there are φ(*Р1*#) subgroups No 4 of mod(*Р1*#), with length R4 ≤ 2*Р<sup>2</sup>* of the whole numbers including one maximal long subgroup No 4 of mod (*Р1*#) with length max R4 = 2*Р<sup>2</sup>* of the whole numbers. At the transition from mod (*Р1*#) to mod(*Р2*#), the fractal *Р1*# and φ(*Р1*#) of groups No 4 repeat *Р<sup>2</sup>* times. Then at the P.I. section from 1 to *Р2*# (at *Р<sup>2</sup>* lines of **Table 1**), we'll get φ(*Р1*#) columns of groups No 4 of mod(*Р1*#), with length R4 ≤ 2*Р2*, (*Р<sup>2</sup>* lines at the column No 4). It is quite obvious that in Section 8.1, it is proved that if by number *Р2*, "eliminate," that is moved to mod(*Р2*#) 1 time every elimination С<sup>2</sup> and С3, in the column of every φ(*Р1*#) group No 4 of mod(*Р1*#), than at the P.I. section from 1 to *Р2*#, (that is at the fractal *Р2*#), we'll get =2φ(*Р1*#) groups No 3 of mod(*Р2*#) of the same length, that is R4 ≤ 2*Р<sup>2</sup>* of mod(*Р1*#) would become = R3 ≤ 2*Р<sup>2</sup>* of mod(*Р2*#) with

As all one by one eliminated residues С<sup>2</sup> or С3, at rearrangement of the groups

**(а)** Fractal = *Р2*#, φ(*Р2*#) group No 3 of mod(*Р2*#), alternance ≤*Р2*, maxR3 = **2***Р2*, with n = (multiple *Р<sup>2</sup>* 1)/*Р1*# = the whole. At the P.I. sections (n*Р1*# *Р2*) and (*Р2*–n)*Р1*# *Р2*. Within the limits of P.I.

> (кр.*Р2*) и С<sup>2</sup> =n*Р1*#. .*1* <sup>2</sup>С и (кр.*Р2*) (*Р2*-n)*Р1*# 1

**(b)** Fractal = *Р3*#, φ(*Р3*#) group No 3 of mod(*Р3*#), alternance ≤*Р3*, maxR3 = **2***Р3*, with n = (multiple

(кр.*Р3*) и С<sup>2</sup> =n*Р2*#. .*1* <sup>2</sup>С и (кр.*Р3*) (*Р3*–n)*Р2*# 1

**(с)** And so on for every mod(*Р*n#), maxR3 = 2*Р*n,n=(кр.*Р*<sup>n</sup> 1)/*Р*n-1# = the whole, *Р*n-(1)-primes

*Type and formula for indexing of two line-symmetrical, maximally long subgroups No 3 (having 3 residues) at*

Within the limits of P.I. section (n*Р2*# *Р4*) with: n and (*Р3*–n) is line number on **Table 2**

3, 5, 7, 3, 11.

3, 5, 7, 3, 11,

… … С<sup>3</sup> … =n*Р1*# + *Р<sup>2</sup>* <sup>3</sup>С = *Р2*#–С<sup>1</sup> (*Р2*–n)*Р1*# + *Р<sup>2</sup>*

… … С<sup>3</sup> … =n*Р2*# + *Р<sup>3</sup>* <sup>3</sup>С = *Р2*#–С<sup>1</sup> (*Р3*–n)*Р2*# + *Р<sup>3</sup>* .. … .СВ… … =n*Р1*# *+ Р<sup>3</sup>* ВС = *Р2*#–СА (*Р2*–n)*Р1*# + *Р<sup>3</sup>*

.. … .СВ… … =n*Р2*# *+ Р<sup>4</sup>* ВС = *Р2*#–СА (*Р3*–n)*Р2*# + *Р<sup>4</sup>*

from No 4 to No3 for the mod(*Р2*#), cannot change the length of none of the subgroups, that is all R3 would permanently be ≤2*Р2*, included in 2φ(*Р1*#) groups No 3 of mod (*Р2*#) there is only one *Р<sup>2</sup>* times repeated, maximally long subgroup No 4 of mod(*Р1*#) with the alternance ≤*Р1*, with length max R4 = 2*Р2*, that would be

from the number = *Р(1)*#/2), that is contrary to the taken axiom.

changing of the alternances composition from ≤*Р<sup>1</sup>* to ≤*Р2*.

section (n*Р1*# *Р3*) with: n and (*Р2*–n) is line number in **Table 1**

11, 3, 7, 5, 3

*Р<sup>3</sup>* 1)/*Р2*# = the whole. At the P.I. sections (n*Р2*# *Р3*) and (*Р3*–n)*Р2*# *Р3*.

11, 3, 7, 5, 3

*the increasing fractal according to the increasing modulus (Tables 11 and 14).*

… ...С<sup>1</sup> … =n*Р1*#*–Р<sup>2</sup>* <sup>1</sup>С = *Р2*#–С<sup>3</sup> (*Р2*–n)*Р1*#–*Р<sup>2</sup>*

… ...С<sup>1</sup> … =n*Р2*#*–Р<sup>3</sup>* <sup>1</sup>С = *Р2*#–С<sup>3</sup> (*Р3*–n)*Р2*#–*Р<sup>3</sup>*

… .СА … . =n*Р1*#*–Р<sup>3</sup>* АС = *Р2*#–СВ (*Р2*–n)*Р1*#–*Р<sup>3</sup>*

...СА … . =n*Р2*#*–Р<sup>4</sup>* АС = *Р3*#–СВ (*Р3*–n)*Р2*#–*Р<sup>4</sup>*

**Table 12.**

**57**

. .

. .

**(with 3 residues) for the mod(***Р2***#)**

*Prime Numbers Distribution Line*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

… С<sup>2</sup> = (*Р1*#–1),С<sup>3</sup> = (*Р1*# + 1) … С<sup>4</sup> = (*Р1*# + *Р2*) … СВ = (*Р1*# + *Р3*). The length of such maximally long subgroup No 4 of mod(*Р1*#), is: max

R4 = (С4–С1)=(*Р1*# + *Р2*)–(*Р1*#–*Р2*)=2*Р<sup>2</sup>* of the whole numbers. The limit of length of the P.I. section within which from СА to СВ would be situated maximal as well as all the other φ(*Р1*#) subgroups No 4 (with 4 residues) of mod(*Р1*#), is: (СВ–СА)=(*Р1*# + *Р3*)–(*Р1*#–*Р3*)=2*Р<sup>3</sup>* of the whole numbers.

It is genuinely:

At the line-symmetrical, primary-repeatable fractal-*Р1*# or I.R.S. according to mod(*Р1*#), φ(*Р1*#) of the least residue (indexed in the form of φ(*Р1*#) groups No 4 of mod(*Р1*#), with the alternances ≤*Р<sup>1</sup>* with different lengths), are situated linesymmetrically relating to the center of symmetry of the fractal-*Р1*#, of the number = (*Р1*#/2). That is they are situated reflecting in pairs and are formed by two different ways: (the left and the right sieve of Eratosthenes), to the left and to the right from the symmetry center of the fractal = *Р1*# of number = *Р1*#/2. To the right – for the increasing values numbers of the P.I. from *Р1*#/2 to *Р1*# to the left for the decreasing values of the P.I. *Р1*#/2 up to 1.

To every left group No 4 of mod(*Р1*#), with the remainder R4 = (С4–С1), is matched by line-symmetrical right group No 4 mod(*Р1*#), with reflecting location of the same first primes number in the same amount and of the same length of the alternance ≤*Р1*: R4 = (*Р1*#–С1)–(*Р1*#–С4)=(С4–С1) consult [7 pp. 142–147, 8 pp. 77–84, 9 pp. 109–116].

Besides two not reflecting that is formed solely subgroup of group No 4: with constant reminder for every *Р<sup>n</sup>* of type: R4 = (*Р1*#/2 + 4)–(*Р1*#/2–4) = 8.

And the section of P.I. fractal *Р1*# (I.R.S. of mod(*Р1*#) from СА to СВ), represented by alternance ≤*Р***<sup>1</sup>** with using of all NOT residues of mod(*Р1*#), (according to the 1 the least > 1), with the subgroup is situated the only maximally long - maximal group No 4 with 4 residues of mod(*Р1*#) (С1-С2-С3-С4). Type: СА = (*Р1*#-*Р3*) … <sup>3</sup>рр<sup>3</sup> … С<sup>1</sup> = (*Р1*#-*Р2*) … <sup>3</sup>рр<sup>3</sup> … С<sup>2</sup> = (*Р1*#-1), С<sup>3</sup> = (*Р1*# + 1) … <sup>3</sup>рр<sup>3</sup> … С<sup>4</sup> = (*Р1*# + *Р2*) … … <sup>3</sup>рр<sup>3</sup> … СВ = (*Р1*# + *Р3*).

Thus in the fractal-*Р1*#-I.R.S. of the mod(*Р1*#), there is only one maximally long subgroup No 4, situated within the maximally long alternance ≤*Р1*, using all NOT residues of mod(*Р1*#), at the P.I. section (*Р1*# *Р2*) with length maximal *R4* = 2*P2* restricting (*R4*–2)/2 = (2*P2*–2)/2 = (*Р2*–1) of the odd numbers, situated within the P. I. section, formed solely from (*Р1*#–*Р3*) to (*Р1*# + *Р3*) with length of (*Р3*–1) of odd numbers.

It is quite obvious that all the other, line-symmetrical subgroups No 4 of mod (*Р1*#), situated within the alternances ≤*Р<sup>1</sup>* with different lengths or NOT residues, of mod(*Р1*#), cannot have the maximal length as they are formed by two different ways, that is they would be shorter than *R4* < 2*P2*, and situated within the P.I. sections from СА to СВ with length not exceeding the maximal long P.I. section (СВ–СА) ≤2*Р<sup>3</sup>* of the whole numbers, not exceeding (*Р3*–1) of the odd numbers.

And so on, for all posterior prime numbers **=** *Р*n, at the increasing fractals -*Р*n# with n - as the whole number and proves the reality of the values of column No 3 of **Table 11** and item (а) of Theorem 1.

It is feasible that there is such a prime number *Р<sup>n</sup>* = *Р(1)*, for which the P.I. is the line-symmetrical fractal *Р(1)*#, situated at the P.I. section from 1 to *Р(1)*# with

#### *Prime Numbers Distribution Line DOI: http://dx.doi.org/10.5772/intechopen.92639*

Then for every recurrent prime number *Р<sup>n</sup>* = *Р<sup>1</sup>* at the P.I., formed as recurrent line symmetrical, primary-repeatable periodical fractal = Р1# or I.R.S. according to mod(Р1#), (see the first line of **Table 1**), the maximal long P.I. section, formed as the alternance of the all first primes number ≤ *Р1*, (NOT residues of mod(*Рn*#)), shall be situated within the P.I. section from СА to СВ, with as the subgroup is the only maximal long maximal subgroup No 4 (С1-С2-С3-С4) with the 4 consequent

… С<sup>2</sup> = (*Р1*#–1),С<sup>3</sup> = (*Р1*# + 1) … С<sup>4</sup> = (*Р1*# + *Р2*) … СВ = (*Р1*# + *Р3*). The length

The limit of length of the P.I. section within which from СА to СВ would be situated maximal as well as all the other φ(*Р1*#) subgroups No 4 (with 4 residues) of

At the line-symmetrical, primary-repeatable fractal-*Р1*# or I.R.S. according to mod(*Р1*#), φ(*Р1*#) of the least residue (indexed in the form of φ(*Р1*#) groups No 4 of mod(*Р1*#), with the alternances ≤*Р<sup>1</sup>* with different lengths), are situated linesymmetrically relating to the center of symmetry of the fractal-*Р1*#, of the number = (*Р1*#/2). That is they are situated reflecting in pairs and are formed by two different ways: (the left and the right sieve of Eratosthenes), to the left and to the right from the symmetry center of the fractal = *Р1*# of number = *Р1*#/2. To the right – for the increasing values numbers of the P.I. from *Р1*#/2 to *Р1*# to the left for

To every left group No 4 of mod(*Р1*#), with the remainder R4 = (С4–С1), is matched by line-symmetrical right group No 4 mod(*Р1*#), with reflecting location of the same first primes number in the same amount and of the same length of the alternance ≤*Р1*: R4 = (*Р1*#–С1)–(*Р1*#–С4)=(С4–С1) consult [7 pp. 142–147,

Besides two not reflecting that is formed solely subgroup of group No 4: with

Thus in the fractal-*Р1*#-I.R.S. of the mod(*Р1*#), there is only one maximally long subgroup No 4, situated within the maximally long alternance ≤*Р1*, using all NOT residues of mod(*Р1*#), at the P.I. section (*Р1*# *Р2*) with length maximal *R4* = 2*P2* restricting (*R4*–2)/2 = (2*P2*–2)/2 = (*Р2*–1) of the odd numbers, situated within the P. I. section, formed solely from (*Р1*#–*Р3*) to (*Р1*# + *Р3*) with length of (*Р3*–1) of odd

It is quite obvious that all the other, line-symmetrical subgroups No 4 of mod (*Р1*#), situated within the alternances ≤*Р<sup>1</sup>* with different lengths or NOT residues, of mod(*Р1*#), cannot have the maximal length as they are formed by two different ways, that is they would be shorter than *R4* < 2*P2*, and situated within the P.I. sections from СА to СВ with length not exceeding the maximal long P.I. section (СВ–СА) ≤2*Р<sup>3</sup>* of the whole numbers, not exceeding (*Р3*–1) of the odd numbers. And so on, for all posterior prime numbers **=** *Р*n, at the increasing fractals -*Р*n# with n - as the whole number and proves the reality of the values of column No 3 of

It is feasible that there is such a prime number *Р<sup>n</sup>* = *Р(1)*, for which the P.I. is the

line-symmetrical fractal *Р(1)*#, situated at the P.I. section from 1 to *Р(1)*# with

constant reminder for every *Р<sup>n</sup>* of type: R4 = (*Р1*#/2 + 4)–(*Р1*#/2–4) = 8. And the section of P.I. fractal *Р1*# (I.R.S. of mod(*Р1*#) from СА to СВ), represented by alternance ≤*Р***<sup>1</sup>** with using of all NOT residues of mod(*Р1*#), (according to the 1 the least > 1), with the subgroup is situated the only maximally long - maximal group No 4 with 4 residues of mod(*Р1*#) (С1-С2-С3-С4). Type: СА = (*Р1*#-*Р3*) … <sup>3</sup>рр<sup>3</sup> … С<sup>1</sup> = (*Р1*#-*Р2*) … <sup>3</sup>рр<sup>3</sup> … С<sup>2</sup> = (*Р1*#-1), С<sup>3</sup> = (*Р1*# + 1) … <sup>3</sup>рр<sup>3</sup> …

mod(*Р1*#), is: (СВ–СА)=(*Р1*# + *Р3*)–(*Р1*#–*Р3*)=2*Р<sup>3</sup>* of the whole numbers.

residue of mod(*Р1*#): Type: СА = (*Р1*#–*Р3*)..С<sup>1</sup> = (*Р1*#–*Р2*).

the decreasing values of the P.I. *Р1*#/2 up to 1.

С<sup>4</sup> = (*Р1*# + *Р2*) … … <sup>3</sup>рр<sup>3</sup> … СВ = (*Р1*# + *Р3*).

**Table 11** and item (а) of Theorem 1.

It is genuinely:

*Number Theory and Its Applications*

8 pp. 77–84, 9 pp. 109–116].

numbers.

**56**

of such maximally long subgroup No 4 of mod(*Р1*#), is: max R4 = (С4–С1)=(*Р1*# + *Р2*)–(*Р1*#–*Р2*)=2*Р<sup>2</sup>* of the whole numbers. subgroup No 4 (containing 4 residues) of mod(*Р(1)*#) with length = R4 > 2\**Р(2)*, situated within the alternance of all first primes number ≤ *Р(1)* within the P.I. section with length > 2\**Р(3),* >(*Р(3)*–1) of the odd numbers. Then every subgroup No 4 would be line-symmetrical to the left and to the right from the center of the fractal *Р(1)*# symmetry of the number *Р(1)*#/2. That is, in the result, we'll get in fractal *Р(1)*# using all primes number ≤ *Р(1)*, we can by more than by one way from the maximally long alternance of all the prime numbers ≤ *Р(1)*, that is by the sieve of Eratosthenes, focused to the left and to the right (to the left and to the right from the number = *Р(1)*#/2), that is contrary to the taken axiom.

#### **6. The maximal length of P.I. section with maximal long subgroups No 3 (with 3 residues) for the mod(***Р2***#)**

At the fractal -*Р1*#, there are φ(*Р1*#) subgroups No 4 of mod(*Р1*#), with length R4 ≤ 2*Р<sup>2</sup>* of the whole numbers including one maximal long subgroup No 4 of mod (*Р1*#) with length max R4 = 2*Р<sup>2</sup>* of the whole numbers. At the transition from mod (*Р1*#) to mod(*Р2*#), the fractal *Р1*# and φ(*Р1*#) of groups No 4 repeat *Р<sup>2</sup>* times. Then at the P.I. section from 1 to *Р2*# (at *Р<sup>2</sup>* lines of **Table 1**), we'll get φ(*Р1*#) columns of groups No 4 of mod(*Р1*#), with length R4 ≤ 2*Р2*, (*Р<sup>2</sup>* lines at the column No 4).

It is quite obvious that in Section 8.1, it is proved that if by number *Р2*, "eliminate," that is moved to mod(*Р2*#) 1 time every elimination С<sup>2</sup> and С3, in the column of every φ(*Р1*#) group No 4 of mod(*Р1*#), than at the P.I. section from 1 to *Р2*#, (that is at the fractal *Р2*#), we'll get =2φ(*Р1*#) groups No 3 of mod(*Р2*#) of the same length, that is R4 ≤ 2*Р<sup>2</sup>* of mod(*Р1*#) would become = R3 ≤ 2*Р<sup>2</sup>* of mod(*Р2*#) with changing of the alternances composition from ≤*Р<sup>1</sup>* to ≤*Р2*.

As all one by one eliminated residues С<sup>2</sup> or С3, at rearrangement of the groups from No 4 to No3 for the mod(*Р2*#), cannot change the length of none of the subgroups, that is all R3 would permanently be ≤2*Р2*, included in 2φ(*Р1*#) groups No 3 of mod (*Р2*#) there is only one *Р<sup>2</sup>* times repeated, maximally long subgroup No 4 of mod(*Р1*#) with the alternance ≤*Р1*, with length max R4 = 2*Р2*, that would be

**(а)** Fractal = *Р2*#, φ(*Р2*#) group No 3 of mod(*Р2*#), alternance ≤*Р2*, maxR3 = **2***Р2*, with n = (multiple *Р<sup>2</sup>* 1)/*Р1*# = the whole. At the P.I. sections (n*Р1*# *Р2*) and (*Р2*–n)*Р1*# *Р2*. Within the limits of P.I. section (n*Р1*# *Р3*) with: n and (*Р2*–n) is line number in **Table 1**


**(b)** Fractal = *Р3*#, φ(*Р3*#) group No 3 of mod(*Р3*#), alternance ≤*Р3*, maxR3 = **2***Р3*, with n = (multiple *Р<sup>3</sup>* 1)/*Р2*# = the whole. At the P.I. sections (n*Р2*# *Р3*) and (*Р3*–n)*Р2*# *Р3*. Within the limits of P.I. section (n*Р2*# *Р4*) with: n and (*Р3*–n) is line number on **Table 2**


**Table 12.**

*Type and formula for indexing of two line-symmetrical, maximally long subgroups No 3 (having 3 residues) at the increasing fractal according to the increasing modulus (Tables 11 and 14).*

**(b)** Fractal = *Р3*#, φ(*Р3*#) group No 2 of mod(*Р3*#), alternance ≤*Р3*, maxR2 = **2***Р2*, with n = (multiple *Р2\*Р<sup>3</sup>* 1)/*Р1*# = the whole. At the P.I. sections (n*Р1*# *Р2*) and (*Р2\*Р3*–n)*Р1*# *Р2*. Within the limits of P.I. section (n*Р1*# *Р3*)., with: n and (*Р2\*Р3*–n) is line number on **Table 1**

**12 3 4**

φ(*Р***0#**)\* \*(Р1–3) R(3) ≤ 2Р<sup>0</sup>

φ(*Р***1#**)\* \*(Р2–3) R(3) ≤ 2Р<sup>1</sup>

φ(*Р***2#**)\* \*(Р3–3) R(3) ≤ 2Р<sup>2</sup>

**…… … …**

Loopback (*Рn*–3) φ(*Рn-1*#) groups No 3 + loopback 2φ(*Рn-1*#) group No 4 [mod*Р*n-1#]= = φ(*Рn*#) groups No 3 of mod(*Рn*#) including 2 groups No 3 = max R3 of mod (*Р*n#) **Proved in Sections 6 and 8**

By repeating *Р<sup>1</sup>* times the line of fractal -*Р0*# and group No 4-3-2, we'll get φ(*Р0*#) columns of groups No 4-3-2 of mod(*Р0*#) within the alternances ≤Р<sup>0</sup> (*Р<sup>1</sup>* line in the column). By "eliminating" 1 number multiple -*Р<sup>1</sup>* (in line of every column No 4-3-2), that is by transiting this group for mod(*Р1*#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р<sup>0</sup> to ≤ Р1. At the P.I. section from 1 to *Р1*# we'll get the fractal -*Р***1#** of mod(*Р***1**#). φ(*Р1*#) groups No 4-3-2

> + 2φ(*Р***0#**) R(4) ≤ 2Р<sup>1</sup>

n = (multiple *Р<sup>1</sup>* 1) / *Р0*# = the whole (n*Р0*#. .*Р1*) and (*Р1*–n)*Р0*#. .*Р<sup>1</sup>* 2 groups No 3- max R3 = 2*Р1*. On the segment of length = (n*Р0*# *Р2*)

By repeating *Р<sup>2</sup>* times the line of fractal –*Р1*# and group No 4-3-2, we'll get φ(*Р1*#) columns of groups No 4-3-2 of mod(*Р1*#) within the alternances ≤Р<sup>1</sup> (*Р<sup>2</sup>* line in the column). By "eliminating" 1 number multiple -*Р<sup>2</sup>* (in line of every column No 4-3-2), that is by transiting this group for mod(*Р2*#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р<sup>1</sup> to ≤ Р2. At the P.I. section from 1 to *Р2*# we'll get the fractal –*Р***2#** of mod(*Р***2**#). φ(*Р2*#) groups No 4-3-2

> + 2φ(*Р***1#**) R(4) ≤ 2Р<sup>2</sup>

n = (multiple *Р<sup>2</sup>* 1) / *Р1*# = the whole (n*Р1*#. .*Р2*) and (*Р2*–n)*Р1*#. .*Р<sup>2</sup>* 2 groups No 3- max R3 = 2*Р2*. On the segment of length = (n*Р1*# *Р3*)

By repeating *Р<sup>3</sup>* times the line of fractal –*Р2*# and group No 4-3-2, we'll get φ(*Р2*#) columns of groups No 4-3-2 of mod(*Р2*#) within the alternances ≤Р<sup>2</sup> (*Р<sup>3</sup>* line in the column). By "eliminating" 1 number multiple–*Р<sup>3</sup>* (in line of every column No 4–3-2), that is by transiting this group for mod(*Р3*#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р<sup>2</sup> to ≤ Р3. At the P.I. section from 1 to *Р3*# we'll get the fractal –*Р***3#** of mod(*Р***3**#). φ(*Р3*#) groups No 4-3-2:

> + 2φ(*Р***2#**) R(4) ≤ 2Р<sup>3</sup>

n = (multiple *Р<sup>3</sup>* 1) / *Р2*# = the whole (n*Р2*#. .*Р3*) and (*Р3*–n)*Р2*#. .*Р<sup>3</sup>* 2 groups No 3- max R3 = 2*Р3*. On the segment of length = (n*Р2*# *Р4*)

And so on for the increasing meanings of modulus = mod *Рn***#**. with: *Р*n**#** - primorial. Р( … ) < Р<sup>0</sup> < Р<sup>1</sup> < Р<sup>2</sup> … < *Р*<sup>n</sup> are the consequent primes. С1-2-3-4 are primes and residues of mod(*Рn***#)** R4-3-2 = (С4–3-2 – С1) that is length of the group = (R4-3-2–2)/2 of odd numbers.

*The loopback of prime's groups and residues rearrangement according to the increasing modulus. With max*

= φ(*Р***1#**) group No 3 R3 ≤ 2Р<sup>1</sup>

= φ(*Р***2#**) group No 3 R3 ≤ 2Р<sup>2</sup>

= φ(*Р***3#**) group No 3 R3 ≤ 2Р<sup>3</sup>

φ(*Р*0#)\* \*(Р1–2) R(2) ≤ 2Р..

φ(*Р*1#)\* \*(Р2–2) R(2) ≤ 2Р<sup>0</sup>

φ(*Р*2#)\* \*(Р3–2) R(2) ≤ 2Р<sup>1</sup>

At every section <2*Р<sup>4</sup>* is group No 3 At every section <2*Р<sup>3</sup>* is group No 2

At every section <2*Р<sup>3</sup>* is group No 3 At every section <2*Р<sup>2</sup>* is group No 2

At every section <2*Р<sup>2</sup>* is group No 3 At every section <2*Р<sup>1</sup>* is group No 2

Loopback (*Рn*–2) φ(*Рn-1*#) Groups No 2 + loopback φ(*Рn-1*#) groups No 3 [mod*Р*n-1#]= =φ(*Рn*#) groups No 2 of mod(*Рn*#) including 2 groups No 2 = max R2 mod (*Р*n#) **Proved in Sections 7 and 9**

> + φ(*Р*0#) R(3) ≤ 2Р<sup>0</sup>

n = (multiple *Р0*\**Р<sup>1</sup>* 1)/*Р(.)*# = the whole (n*Р(.)*# *Р0*) and (*Р0Р1*–n)*Р(.)*# *Р<sup>0</sup>* 2 groups No2- max R2 = 2*Р0*. On the segment of length = (n*Р*(..)# *Р1*)

> + φ(*Р*1#) R(3) ≤ 2Р<sup>1</sup>

n = (multiple *Р2*\**Р<sup>1</sup>* 1)/*Р0*# = the whole (n*Р0*# *Р1*) and (*Р2Р1*–n)*Р0*# *Р<sup>1</sup>* 2 groups No2- max R2 = 2*Р1*. On the segment of length = (n*Р*0# *Р2*)

> + φ(*Р*2#) R(3) ≤ 2Р<sup>2</sup>

n = (multiple *Р2*\**Р<sup>3</sup>* 1)/*Р1*# = the whole (n*Р1*# *Р2*) and (*Р2Р3*–n)*Р1*# *Р<sup>2</sup>* 2 groups No 2- max R2 = 2*Р2*. On the segment of length = (n*Р*1# *Р3*)

= φ(*Р*1#) group No 2 R2 ≤ 2Р<sup>0</sup>

= φ(*Р*2#) group No 2 R2 ≤ 2Р<sup>1</sup>

= φ(*Р*3#) group No 2 R2 ≤ 2Р<sup>2</sup>

φ(*Рn*#) groups No 4 of mod (*Рn*#). Including 1 group = maxR4 **Proof in Section 5**

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

*Prime Numbers Distribution Line*

φ(*Р***1#**) groups No 4. R4. ≤ 2Р<sup>2</sup> at every sectionе<2*Р<sup>3</sup>* is group No 4

(*Р1*# *Р2*) max R4 = 2Р<sup>2</sup> At the section with length = (*Р1*# *Р3*)

φ(*Р***2#**) groups No 4. R4. ≤ 2Р<sup>3</sup> at every sectionе < 2*Р<sup>4</sup>* is group No 4

(*Р2*# *Р3*) max R4 = 2Р<sup>3</sup> At the section with length = (*Р2*# *Р4*)

φ(*Р***3#**) groups No 4. R4. ≤ 2Р<sup>4</sup> at every section <2*Р<sup>5</sup>* is group No 4

(*Р3*# *Р4*) max R4 = 2Р<sup>4</sup> At the section with length = (*Р3*# *Р5*)

*R = const from mod(*Р*1#) to mod(*Р*3#).*

Fractal composition - *Рn*#

Amount of groups of Length and location

Formula = n Type max R Groups and location

Amount of groups of length and location

Formula = n Type max R Groups and location

Amount of groups of length and location

Formula = n Type max R Groups and location

**Table 14.**

**59**


**(c)** Fractal = *Р4*#, φ(*Р4*#) group No 2 of mod(*Р4*#), alternance ≤*Р4*, maxR2 = **2***Р3*, with n = (multiple *Р4\*Р<sup>3</sup>* 1)/*Р2*# = the whole. At the P.I. sections (n*Р2*# *Р3*) and (*Р4\*Р3*–n)*Р2*# *Р3*. Within the limits of P.I. section (n*Р2*# *Р4*)., with: n-and (*Р4\*Р3*–n) is line number on **Table 2**


#### **Table 13.**

*Type and formula for indexing of two line-symmetrical, maximally long subgroups No 2 (having 2 residues) at the increasing fractal according to the increasing modulus (Tables 11 and 14).*

restructured into two maximally long line-symmetrical groups No 3 of mod(*Р2*#) by "eliminating" the residues С<sup>2</sup> and С<sup>3</sup> by number multiple to *Р2*, (1 time at *Р<sup>2</sup>* lines). As all the other 2φ(*Р1*#)–2 subgroups, changed from No 4 to No 3 for mod(*Р2*#) are shorter than (*Р2*–1) of the odd numbers that is: R3 < 2*Р*2. In Sections 8.1 and 8.3, there are no other ways of making or changing the subgroups No 3 of mod(*Р2*#) with length R3 > 2*Р*2.

Order, type, and formula of indexing of two subgroups No 3 according to the increasing modulus are represented in **Table 12** а, b, с.

The length of these two line-symmetrical subgroups No 3 of mod(*Р2*#), that is the length of alternance ≤*Р<sup>2</sup>* from С<sup>1</sup> to С3, is maximal R3 = (С3–С1) = (n*Р1*#. *Р2*)– (n*Р1*#.–*Р2*) =2*Р2*; (*Р2*–1) of the odd numbers. Two of these subgroups No 3 are situated within P.I. section (n*Р1*#. *Р3*) with length (С*В*–СА)=2*Р3*; (*Р3*–1) of the odd numbers from СА to СВ. Numerical values of these two maximal subgroups No 3 are defined according to the formula (multiple *Р<sup>2</sup>* and С<sup>2</sup> = multiple *Р<sup>2</sup>* 2) is (n*Р1*#. .1) and (*Р2*–n)*Р1*# 1, with n and (*Р2*–n) define the number of the line for the group No 3 of mod(*Р2*#) in column *Р<sup>2</sup>* and the repeated maximal of the group No 4 of mod(*Р1*#) with the period = *Р1*# (consult **Table 1**). That is n = (multiple *Р2*. 1)/*Р1*# = the whole < *Р2*/2.

Whereas, it is quite obvious that in and proved in Section 8.2, the other subgroups No 3 of mod(*Р2*#), with different lengths, changed from groups No 4 would be within P.I. section, with limit length = 2*Р<sup>3</sup>* of the whole numbers.

And so on, for every of all posterior primes **=** *Р*n, at the increasing fractals *Р*n#, with n is the whole, represented in **Tables 11** and **14** (the proof is indicated in Section 10). (Numerical illustrations are in **Table 15**).

#### **7. The maximal length of the P.I. section, where two maximally long subgroups No 2 (with 2 residues) for the mod(***Р3***#)**

Representing as one line, the first *Р<sup>2</sup>* lines in **Table 1** we'll get the fractal *Р2*# according to the mod(*Р2*#) – I.R.S. at mod(*Р2*#), that is situated at the P.I. section *Prime Numbers Distribution Line DOI: http://dx.doi.org/10.5772/intechopen.92639*


**Table 14.**

*The loopback of prime's groups and residues rearrangement according to the increasing modulus. With max R = const from mod(*Р*1#) to mod(*Р*3#).*

restructured into two maximally long line-symmetrical groups No 3 of mod(*Р2*#) by "eliminating" the residues С<sup>2</sup> and С<sup>3</sup> by number multiple to *Р2*, (1 time at *Р<sup>2</sup>* lines). As all the other 2φ(*Р1*#)–2 subgroups, changed from No 4 to No 3 for mod(*Р2*#) are shorter than (*Р2*–1) of the odd numbers that is: R3 < 2*Р*2. In Sections 8.1 and 8.3, there are no other ways of making or changing the subgroups No 3 of mod(*Р2*#)

**(b)** Fractal = *Р3*#, φ(*Р3*#) group No 2 of mod(*Р3*#), alternance ≤*Р3*, maxR2 = **2***Р2*, with n = (multiple

(кр.*Р2*) (кр.*Р3*) =n*Р1*#. .*1* (кр.*Р3*) (кр.*Р2*) (*Р2Р3*–n)*Р1*# 1

**(c)** Fractal = *Р4*#, φ(*Р4*#) group No 2 of mod(*Р4*#), alternance ≤*Р4*, maxR2 = **2***Р3*, with n = (multiple

(кр.*Р4*) (кр.*Р3*) =n*Р2*#. .*1* (кр.*Р3*) (кр.*Р4*) (*Р4Р3*– n)*Р2*# 1

**(d)** And so on for every mod(*Р*n+1#), maxR2 = 2*Р*n,n=(*Рn + 1\*Р*<sup>n</sup> 1)/*Р*n-1# = the whole, *Р*n-primes

*Type and formula for indexing of two line-symmetrical, maximally long subgroups No 2 (having 2 residues) at*

Within the limits of P.I. section (n*Р2*# *Р4*)., with: n-and (*Р4\*Р3*–n) is line number on **Table 2**

3, 5, 7. .. ..

3, 5, 7. .. ..

… … С<sup>2</sup> … =n*Р1*# + *Р<sup>2</sup>* <sup>2</sup>С = *Р3*#–С<sup>1</sup> (*Р2Р3*– n)*Р1*# + *Р<sup>2</sup>*

… … С<sup>2</sup> … =n*Р2*# + *Р<sup>3</sup>* <sup>2</sup>С = *Р4*#–С<sup>1</sup> (*Р4Р3*– n)*Р2*# + *Р<sup>3</sup>*

. .

. .

… .СВ … … =n*Р1*# *+ Р<sup>3</sup>* ВС = *Р3*#–СА (*Р2Р3*–n)*Р1*# + *Р<sup>3</sup>*

> … .СВ … … =n*Р2*# *+ Р<sup>4</sup>* ВС = *Р4*#–СА (*Р4Р3*– n)*Р2*# + *Р<sup>4</sup>*

*Р2\*Р<sup>3</sup>* 1)/*Р1*# = the whole. At the P.I. sections (n*Р1*# *Р2*) and (*Р2\*Р3*–n)*Р1*# *Р2*. Within the limits of P.I. section (n*Р1*# *Р3*)., with: n and (*Р2\*Р3*–n) is line number on **Table 1**

*Р4\*Р<sup>3</sup>* 1)/*Р2*# = the whole. At the P.I. sections (n*Р2*# *Р3*) and (*Р4\*Р3*–n)*Р2*# *Р3*.

.. .. 7, 5, 3

*the increasing fractal according to the increasing modulus (Tables 11 and 14).*

.. .. 7, 5, 3

Order, type, and formula of indexing of two subgroups No 3 according to the

The length of these two line-symmetrical subgroups No 3 of mod(*Р2*#), that is the length of alternance ≤*Р<sup>2</sup>* from С<sup>1</sup> to С3, is maximal R3 = (С3–С1) = (n*Р1*#. *Р2*)– (n*Р1*#.–*Р2*) =2*Р2*; (*Р2*–1) of the odd numbers. Two of these subgroups No 3 are situated within P.I. section (n*Р1*#. *Р3*) with length (С*В*–СА)=2*Р3*; (*Р3*–1) of the odd numbers from СА to СВ. Numerical values of these two maximal subgroups No 3 are defined according to the formula (multiple *Р<sup>2</sup>* and С<sup>2</sup> = multiple *Р<sup>2</sup>* 2) is (n*Р1*#. .1) and (*Р2*–n)*Р1*# 1, with n and (*Р2*–n) define the number of the line for the group No 3 of mod(*Р2*#) in column *Р<sup>2</sup>* and the repeated maximal of the group No 4 of mod(*Р1*#) with the period = *Р1*# (consult **Table 1**). That is n = (multiple

Whereas, it is quite obvious that in and proved in Section 8.2, the other subgroups No 3 of mod(*Р2*#), with different lengths, changed from groups No 4 would

And so on, for every of all posterior primes **=** *Р*n, at the increasing fractals *Р*n#, with n is the whole, represented in **Tables 11** and **14** (the proof is indicated in

**7. The maximal length of the P.I. section, where two maximally long**

Representing as one line, the first *Р<sup>2</sup>* lines in **Table 1** we'll get the fractal *Р2*# according to the mod(*Р2*#) – I.R.S. at mod(*Р2*#), that is situated at the P.I. section

be within P.I. section, with limit length = 2*Р<sup>3</sup>* of the whole numbers.

**subgroups No 2 (with 2 residues) for the mod(***Р3***#)**

Section 10). (Numerical illustrations are in **Table 15**).

increasing modulus are represented in **Table 12** а, b, с.

with length R3 > 2*Р*2.

… ...СА … . =n*Р1*#*–Р<sup>3</sup>* АС = *Р3*#–СВ (*Р2Р3*–n)*Р1*#–

… ...СА … . =n*Р2*#*–Р<sup>4</sup>* АС = *Р4*#–СВ (*Р4Р3*–n)*Р2*#– . .

*Number Theory and Its Applications*

. .

… ...С<sup>1</sup> … =n*Р1*#*–Р<sup>2</sup>* <sup>1</sup>С = *Р3*#–С<sup>3</sup> (*Р2Р3*–n)*Р1*#– *Р2*

… ...С<sup>1</sup> … =n*Р2*#*–Р<sup>3</sup>* <sup>1</sup>С = *Р4*#–С<sup>3</sup> (*Р4Р3*–n)*Р2*#– *Р3*

*Р3*

*Р4*

**Table 13.**

**58**

*Р2*. 1)/*Р1*# = the whole < *Р2*/2.

from 1 to *Р2*# and represented in 1 line of **Table 2** where φ(*Р2*#) of groups No 3 of mod(*Р2*#) are situated with length R3 ≤ 2*Р<sup>2</sup>* of the whole numbers. In this number, the two maximally long groups No 3 of mod(*Р2*#) with length max R3 = 2*Р<sup>2</sup>* of the whole numbers are represented. Then on the P.I. section from 1 to *Р3*# (at *Р<sup>3</sup>* lines of **Table 2**), we'll get φ(*Р2*#) columns of group No 3 of mod(*Р2*#), with length R3 ≤ 2*Р2*, (*Р<sup>3</sup>* lines in columns of groups No 3).

It is quite obvious, that in Section 9.1, it is proved that if by number *Р3*, "eliminate," that is to change for the model mod(*Р3*#) residue С<sup>2</sup> in the column of every φ(*Р2*#) group No 3 of mod(*Р2*#), then at the P.I. section from 1 to *Р3*#, that is at the fractal *Р3*#, we'll get φ(*Р2*#) groups No 2 of mod(*Р3*#) of the same length, that is R3 ≤ 2*Р<sup>2</sup>* mod(*Р2*#), would become R2 ≤ 2*Р<sup>2</sup>* of mod(*Р3*#) with changing the structure of alternance from ≤*Р<sup>2</sup>* to ≤*Р3*.

As any eliminated residue С2, during the rearrangement of groups from No 3 to No 2 for the mod(*Р3*#) cannot change the length on no subgroup that is all R2 would permanently ≤2*Р2*, and included in φ(*Р2*#) groups No 3 of mod(*Р2*#) there are two uncial, repeated *Р<sup>3</sup>* times maximally long subgroups No 3 of mod(*Р2*#) with the alternance ≤*Р<sup>2</sup>* with length maximal R3 = 2*Р2*, that would be rearranged into two maximally long line-symmetrical groups No 2 of mod(*Р3*#) by "eliminating" the residues С<sup>2</sup> with the number multiple *Р3*, (1 time in *Р<sup>3</sup>* lines). As all the other φ(*Р2*#)–2 subgroups, rearranged from No 3 to No 2 for the mod(*Р3*#), are shorter than (*Р2*–1) of the off numbers, that is: R2 < 2*Р*2, and in Sections 9.1 and 9.3, it is proved that there are no other ways of comparing or rearranging of the subgroups No 2 of mod(*Р2*#) with length R3 > 2*Р*2. The order, type, and formula of indexing of two subgroups No 2 according to the increasing modulus are represented in **Table 13** b, с, d.

The length of these two line-symmetrical subgroups No 2 of mod(*Р3*#), the length of alternance ≤*Р3*, from С<sup>1</sup> to С2, that is max R2 = (С2–С1) = (n*Р1*# + *Р2*)– (n*Р1*#–*Р2*) =2*Р2*; (*Р2*–1) of odd numbers. Two of these subgroups No 2 are situated within the P.I. section (n*Р1*#. *Р3*) with length (С*В*–СА)=2*Р3*; (*Р3*–1) of odd numbers from СА to СВ. The numerical values of these maximal groups No 2 is defined according to the formula: (multiple *Р<sup>2</sup>* and multiple *Р3*) is (n*Р1*#. .1) and (*Р2Р3*–n)*Р1*#. .1., with n and (*Р2Р3*–n) define the line number for the group No 2 of mod(*Р3*#) in the column -*Р2*\**Р<sup>3</sup>* of the duplication of the max group No 4 of mod (*Р1*#) with the period = *Р1*# (**Table 1**). That is n = (multiple *Р2*\**Р3*. 1)/*Р1*# = the whole < *Р2\*Р3*/2.

Herewith, it is quite obvious, and proved in Section 9.3, that all the other subgroups No 2 of mod(*Р3*#), with different lengths, rearranges from groups No 3 would be within the P.I., with length not exceeding the limit =2*Р<sup>3</sup>* of the wholes.

And so on, for every of all posterior primes **=** *Р*n, at the increasing fractals *Р*n#, with n is the whole, represented in **Tables 11** and **14** (the proof is indicated in Section 10) (numerical illustrations are in **Table 16**).

**for the fractal**

> **СА = 79**

**61**

**…**

**С1 = 83.**

**5,3**

**С2 = 89 и (7\*13)**

**3,5**

**С3 = 97**

**..**

**СВ = 101**

**ВС = 131**

**3С = 127**

С*В*–СА) = 2\*13, n = 1

**(7\*17) и 2С = 121**

**С1 = 113** for the fractal

*11*# = 2310, mod(*11*#), maxR3 = 2\*11, at the P.I. section (

**АС = 109** .СА = 197.

…

..С1 = 199.

3,7,

(11\*19),

=n7#.

2С =

2099,(11\*191)

=(11–n)7#

 1

 .*<sup>1</sup>*

С2 = 211

3,5,

..С3 = 221

=n7# + 11

… 3С = 2111

(11–n)7# + 11

С*В*–СА) = 2\*17, n = 3

…

..

.СВ = 223

*Prime Numbers Distribution Line*

=n7# + 13 ВС = 2113.

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

(11–n)7# + 13

7,3,

5,3

=n7#*–11*

1С = 2089.

=11#–С3

for the fractal

*13*# = 30,030, mod(*13*#), maxR3 = 2\*13, at the P.I. section (

=n7#*–13*

.АС = 2087.

=11#–СВ

.СА = 6913.

.

..С1 = 6917.

11,

(13\*533),

=n11#.

 .*<sup>1</sup>*

С2 = 6931

3,5,

С3 = 6943

..

.СВ = 6947

=n11# + 17 ВС = 23,117. (13–n)11# + 17

=n11# + 13

… 3С = 23,113 (13–n)11# + 13

С*В*–СА) = 2\*19,n = 2

7,3,

11,

3,7,

5,3

2С =

23,099,(13\*1777)

=(13–n)11#

 1

=n11#*–13*

1С = 23,087.

=13#–С3

for the fractal

*17*# =

510,510,mod(*17*#),

 maxR3 = 2\*17, at the P.I. section (

…

=n11#*–17*

.АС = 23,083.

=13#–СВ

.СА = 60,041.

.

..С1 = 60,043.

13,

С2 =

60,059,(17\*3533)

3,5,

С3 = 60,077..

..

.СВ = 60,079

=n13# + 19

ВС = 450,469 (17–n)13# + 19

=n13# + 17

7,3,

11,

… 3С = 450,467 (17-n)13# + 17

С*В –*СА) = 2\*23, n = 1

13,

=n13#.

 .*<sup>1</sup>*

11,

3,7,

(17\*26497), 2С = 450,451

=(17–n)13#

 1

5,3

=n13#*–17*

1С = 450,433.

=17#–С3

for the fractal

*19*# =

9,699,690,mod(*19*#),

 maxR3 = 2\*19, at the P.I. section (

…

=n13#*–19*

.АС = 450,431.

=17#–СВ

СА = 510,487

…

С1 = 510,491

17..

С2 = 510,509 и кр.19.

3,5,

..С3 = 510,529 … 3С = 9,189,199

 ..

 .

ВС = 9,189,203

СВ = 510,533

...17,

5,3

Кр.19 и 2С = 9,189,181

1С = 9,189,161

АС = 9,189,157

**Table 15.** *The numerical*

 *examples of the two* 

*line-symmetrical*

 *maximally*

 *long subgroups No 3 (containing*

 *3 residues =*

*С1-2-3), according to the increasing modulus.*

*7***# = 210, mod(7#), maxR3 = 2\*7, at the P.I. section (**

**С***В***–СА) = 2\*11, n = 3**

#### **8. The loopback of rearrangement for φ(***Р***n#) groups No 3 (with 3 residues) from mod(***Р***n-1#) for mod(***Р***n#)**

The loopback order of rearrangement of φ(*Р*n#) groups No 3, according to the increasing modulus, that are represented in column No 3 of **Table 14** at Section 8 are examined by steps, for every recurrent increasing fractal -*Р*n#:

During the transition from mod(*Р1*#) to mod(*Р2*#) of the fractal -*Р1*# (1 line of **Table 1**) and every from φ(*Р1*#) groups No 4-3-2 of mod(*Р1*#) are repeated *Р<sup>2</sup>* times. That at the P.I. section from 1 to *Р2*#, we'll get φ(*Р1*#) columns of No 4-3-2 groups (*Р<sup>2</sup>* limes at the column). Number *Р<sup>2</sup>* according to diagonals of *Р<sup>2</sup>* lines


#### **15.** *numerical examples of the two line-symmetrical maximally long subgroups No 3 (containing 3 residues = С1-2-3), according to the increasing modulus.*

**Table** 

*The* 

#### *Prime Numbers Distribution Line DOI: http://dx.doi.org/10.5772/intechopen.92639*

from 1 to *Р2*# and represented in 1 line of **Table 2** where φ(*Р2*#) of groups No 3 of mod(*Р2*#) are situated with length R3 ≤ 2*Р<sup>2</sup>* of the whole numbers. In this number, the two maximally long groups No 3 of mod(*Р2*#) with length max R3 = 2*Р<sup>2</sup>* of the whole numbers are represented. Then on the P.I. section from 1 to *Р3*# (at *Р<sup>3</sup>* lines of **Table 2**), we'll get φ(*Р2*#) columns of group No 3 of mod(*Р2*#), with length

It is quite obvious, that in Section 9.1, it is proved that if by number *Р3*, "eliminate," that is to change for the model mod(*Р3*#) residue С<sup>2</sup> in the column of every φ(*Р2*#) group No 3 of mod(*Р2*#), then at the P.I. section from 1 to *Р3*#, that is at the fractal *Р3*#, we'll get φ(*Р2*#) groups No 2 of mod(*Р3*#) of the same length, that is R3 ≤ 2*Р<sup>2</sup>* mod(*Р2*#), would become R2 ≤ 2*Р<sup>2</sup>* of mod(*Р3*#) with changing the

As any eliminated residue С2, during the rearrangement of groups from No 3 to No 2 for the mod(*Р3*#) cannot change the length on no subgroup that is all R2 would permanently ≤2*Р2*, and included in φ(*Р2*#) groups No 3 of mod(*Р2*#) there are two uncial, repeated *Р<sup>3</sup>* times maximally long subgroups No 3 of mod(*Р2*#) with the alternance ≤*Р<sup>2</sup>* with length maximal R3 = 2*Р2*, that would be rearranged into two maximally long line-symmetrical groups No 2 of mod(*Р3*#) by "eliminating" the residues С<sup>2</sup> with the number multiple *Р3*, (1 time in *Р<sup>3</sup>* lines). As all the other φ(*Р2*#)–2 subgroups, rearranged from No 3 to No 2 for the mod(*Р3*#), are shorter than (*Р2*–1) of the off numbers, that is: R2 < 2*Р*2, and in Sections 9.1 and 9.3, it is proved that there are no other ways of comparing or rearranging of the subgroups No 2 of mod(*Р2*#) with length R3 > 2*Р*2. The order, type, and formula of indexing of

two subgroups No 2 according to the increasing modulus are represented in

The length of these two line-symmetrical subgroups No 2 of mod(*Р3*#), the length of alternance ≤*Р3*, from С<sup>1</sup> to С2, that is max R2 = (С2–С1) = (n*Р1*# + *Р2*)– (n*Р1*#–*Р2*) =2*Р2*; (*Р2*–1) of odd numbers. Two of these subgroups No 2 are situated within the P.I. section (n*Р1*#. *Р3*) with length (С*В*–СА)=2*Р3*; (*Р3*–1) of odd numbers from СА to СВ. The numerical values of these maximal groups No 2 is defined according to the formula: (multiple *Р<sup>2</sup>* and multiple *Р3*) is (n*Р1*#. .1) and (*Р2Р3*–n)*Р1*#. .1., with n and (*Р2Р3*–n) define the line number for the group No 2 of mod(*Р3*#) in the column -*Р2*\**Р<sup>3</sup>* of the duplication of the max group No 4 of mod (*Р1*#) with the period = *Р1*# (**Table 1**). That is n = (multiple *Р2*\**Р3*. 1)/*Р1*# = the

Herewith, it is quite obvious, and proved in Section 9.3, that all the other subgroups No 2 of mod(*Р3*#), with different lengths, rearranges from groups No 3 would be within the P.I., with length not exceeding the limit =2*Р<sup>3</sup>* of the wholes. And so on, for every of all posterior primes **=** *Р*n, at the increasing fractals *Р*n#, with n is the whole, represented in **Tables 11** and **14** (the proof is indicated in

**8. The loopback of rearrangement for φ(***Р***n#) groups No 3 (with 3**

are examined by steps, for every recurrent increasing fractal -*Р*n#:

The loopback order of rearrangement of φ(*Р*n#) groups No 3, according to the increasing modulus, that are represented in column No 3 of **Table 14** at Section 8

During the transition from mod(*Р1*#) to mod(*Р2*#) of the fractal -*Р1*# (1 line of **Table 1**) and every from φ(*Р1*#) groups No 4-3-2 of mod(*Р1*#) are repeated *Р<sup>2</sup>* times. That at the P.I. section from 1 to *Р2*#, we'll get φ(*Р1*#) columns of No 4-3-2 groups (*Р<sup>2</sup>* limes at the column). Number *Р<sup>2</sup>* according to diagonals of *Р<sup>2</sup>* lines

Section 10) (numerical illustrations are in **Table 16**).

**residues) from mod(***Р***n-1#) for mod(***Р***n#)**

R3 ≤ 2*Р2*, (*Р<sup>3</sup>* lines in columns of groups No 3).

*Number Theory and Its Applications*

structure of alternance from ≤*Р<sup>2</sup>* to ≤*Р3*.

**Table 13** b, с, d.

whole < *Р2\*Р3*/2.

**60**

"eliminates," that is rearranges to the mod(*Р2*#) one time every of Р<sup>2</sup> repeated numbers of the P.I. section from 1 to Р1# (1 number at every Р<sup>2</sup> line of every column No 4 and No 3), "eliminating" the residues С1-С2-С3-С<sup>4</sup> in the groups No 4 (consult Section 8.1), and ALL numbers, besides the residues С1-С2-С<sup>3</sup> in the groups No 3 (consult Section 8.3).

mod(*Р2*#) with "changing" the alternance from ≤*Р<sup>1</sup>* to ≤*Р2*. Without changes the groups No 3 length for the mod(*Р2*#), *R3* = const and numbers composition within the alternances ≤*Р2*; that is the previously "eliminated" ≤ *Р1*, according to the 1 least

Total at the fractal -*Р2*# at the P.I. section from 1 to *Р2*# we'll get the loopback of the rearranged groups No 3 for the mod(*Р2*#) represented in Sections

That is: 2φ(Р1#) (Section 8.1 with the length = R3 ≤ 2Р2) + φ(Р1#)(Р2–3) (Section 8.3 with the length R3 < < R4 = 2Р2) = Р2φ(Р1#)–3φ(Р1#) + 2φ(Р1#) = *Р<sup>2</sup>* φ(*Р1*#)– φ(*Р1*#) = φ(*Р1*#)(*Р2*–1) = φ(*Р2*#) of the subgroups No 3 for the mod(*Р2*#), included at the alternances ≤*Р<sup>2</sup>* with length R3 ≤ 2*Р*2, including two maximally long sub-

As all φ(*Р***2**#) of the subgroups No 3 of mod(*Р2*#) are examined in Sections 8.1 and 8.3, with all eliminated one time residues С<sup>2</sup> or С3, examined in Section 8.1, cannot change the length = R3 ≤ 2*Р2*, none of the 2φ(*Р1*#) groups, rearranged from No 4 to No 3 for the mod(*Р2*#). Herewith the residues С*<sup>1</sup>* and С*<sup>4</sup>* are also accounted in two adjoining groups of **Table 17** as С*<sup>2</sup>* or С*3*. And indiscriminately φ(*Р1*#)(*Р2*–3) groups No 3 for the mod(*Р2*#), examined in Section 8.3 are shorter than limit

So, there are no other ways to make groups No 3 of mod(*Р2*#) with length R3 > 2*Р2,* besides the way to form the maximally long subgroups No 3 with length

Thus we got, that for every recurrent = *Р2*, φ(*Р2*#) residues of mod(*Р2*#) situated in the fractal -*Р2*# (*Р<sup>2</sup>* lines of **Table 1**), represented as loop back φ(*Р2*#) of subgroups No 3 (3 residues of mod(*Р2*#), represented in Section 8.4), that is pure

mod(*Р2*#) at every P.I. section, with length not exceeding 2*Р<sup>3</sup>* of the whole numbers

And so on, for every of all eventual primes **=** *Р*n, represented as the loopback of groups distribution of the residues No 3 at the increasing fractals -*Р*n# according to the increasing meanings of modulus-mod(*Р*n#) that proves the validity of section (b) of the Theorem 1 (loopback of groups No 3 is represented in column No 3 of

**9. The loopback of rearrangement for φ(***Р***n#) groups No 2 (2 residues)**

The looped back order of rearrangement φ(*Р*n#) of No 2 groups, according to the increasing modulus, that are represented in column No 4 of **Table 14**, in Section 9

Representing the first *Р<sup>2</sup>* lines in **Table 1** as one line, we'll get the fractal -*Р2*# according to mod(*Р2*#)-I.R.S of mod(*Р2*#) at the P.I. sections from 1 to *Р2*# (1 line

With φ(*Р2*#) line-symmetrical the least residues of mod(*Р2*#), which according to Section 2 are indexed according to φ(*Р2*#) groups of residues No 4-3-2 for the

*<sup>2</sup>* at the P.I. section

*<sup>2</sup>* to *Р2*#, pure THREE consequent residue of

>1 from the number are accounted. Type: С<sup>1</sup> ррС<sup>2</sup> ррС3.

max R3 = 2*Р2*, represented in Sections 6 and 8.1.

. And further, from *Р<sup>3</sup>*

R3 = (CС–AС) ≤ **2***Р<sup>2</sup>* (consult Sections 8.1 and 8.4).

**from the mod(***Р***n-1#) to mod(***Р***n#)**

THREE consequent prime (A,B,CС) type: *P2 <* (AС-BС*-*CС) < *P3*

are examined by steps for every recurrent increasing fractal -*Р*n#:

(see Section 8.2), with length of every subgroup No 3 at every section is

8.4

8.1 and 8.3.

8.5

R3 = 2*Р2.*

8.6

from *Р<sup>2</sup>* to *Р<sup>3</sup>*

**Table 14**).

of **Table 2**).

mod(*Р2*#).

**63**

*2*

groups No 3 max R3 = 2*Р*2.

*Prime Numbers Distribution Line*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

With: *R3* = const =? (with to mod(*Р2*#) R3 *< <* R4 = 2*Р*2).

8.1

It is quite obvious, that after "elimination" from every φ(*Р1*#) column of the group No 4 of mod(*Р1*#) one time every residue С<sup>2</sup> and С3, we'll get at *Р<sup>2</sup>* lines of every column of the No 4 groups of mod(*Р1*#), –TWO groups No 3 of mod(*Р2*#)

That is, we'll get 2φ(*Р1*#) subgroups No 3 of mod(*Р2*#) with invariance length of the previous groups, that is *R4* φ(*Р1*#) groups of No 4 mod(*Р1*#), would become = *R3* for 2φ(*Р1*#) groups No 3 of mod(*Р2*#), with changing the alternance structure from ≤*Р<sup>1</sup>* to ≤*Р2*, that are situated at the P.I section from 1 to *Р2*# that is at the fractal-*Р2*#.

Herewith within the 2φ(*Р1*#) groups No 3 of mod(*Р2*#), there are accounted all residues = С<sup>1</sup> and = С<sup>4</sup> of mod(*Р1*#) and alternances ≤*Р<sup>2</sup>* of such rearranged groups from No 4 of mod(*Р1*#) to No 3 of mod(*Р2*#). As along of *Р<sup>2</sup>* duplication of the three adjoined groups No 4 of mod(*Р1*#), represented in **Table 3**, we'll get **Table 17**, with every residue = С<sup>2</sup> or = С3, situated at one of the 3 lines of **Table 3** (for example, at line а of **Table 17**), is accounted as residue С<sup>1</sup> or С<sup>4</sup> at the other two adjoined groups No 4 (at lines: b or с of **Table 17**), where they are situated within 4 consequent residues, going as the second or the third, that is they are "excluded" as С2 or 3 in these adjoined groups (lines):


Thus, "eliminating" that means transition to mod(*Р2*#), one time every 4 residue in φ(*Р1*#) groups No 4 of mod(*Р1*#), at the P.I. sections from 1 to *Р2*#, that is at the fractal -*Р2*#, we'll, get the loopback, represented as 2φ(*Р1*#) groups No 3 of mod (*Р2*#) type: С<sup>1</sup> ррС<sup>2</sup> рр(С2–<sup>3</sup> = multiple *Р2*) ррС3, with the alternances ≤*Р2*, length *R3* ≤ *2Р2*. Including, pure 2 subgroups No 3 of mod(*Р2*#) with length maximal R3 = 2*Р*2.

#### 8.2

Herewith, it is quite obvious that every three consequent residues of every subgroup No 3 according to the increasing group of mod(Рn#), represented in **Table 17**, are still within the P.I. section with length not exceeding - 2*Р*n+1 of the whole numbers, as "eliminated" residues С2;3 and С1;4 of mod(Рn#) doesn't change the location of every subgroup No 3. That is, we'll get at the three adjoined groups No 3 lines of **Table 17** for the mod(*Р2*#): type (а) =(СВ–СА) *< 2Р3*; type (b) = (С2–С1) *< 2Р3*; type (с) =(С4–С3) *< 23*.

Including pure two subgroups No 3 max R3 = 2*Р*2, located within the maximally long section with length = 2*Р<sup>3</sup>* of the whole numbers. (The rearrangement is studied in Section 6).

8.3

Within the *Р<sup>2</sup>* duplications φ(*Р1*#) of the groups No 3 of mod(*Р1*#), number *Р<sup>2</sup>* "eliminated," that is rearranges to the mod(*Р2*#) 1 time every of all previously eliminated numbers of group No 3, except three residues С1–С2–С3. That is, transits to the mod(*Р2*#) (*Р2*–3) of No 3 groups in every φ(*Р1*#) column of No 3 groups.

Then at the P.I. section from 1 to *Р2*#, (that is included into fractal -*Р2*#), we'll get the loop back, represented as (*Р2*–3) φ(*Р1*#) of No 3 groups repetition for the

mod(*Р2*#) with "changing" the alternance from ≤*Р<sup>1</sup>* to ≤*Р2*. Without changes the groups No 3 length for the mod(*Р2*#), *R3* = const and numbers composition within the alternances ≤*Р2*; that is the previously "eliminated" ≤ *Р1*, according to the 1 least >1 from the number are accounted. Type: С<sup>1</sup> ррС<sup>2</sup> ррС3.

With: *R3* = const =? (with to mod(*Р2*#) R3 *< <* R4 = 2*Р*2). 8.4

Total at the fractal -*Р2*# at the P.I. section from 1 to *Р2*# we'll get the loopback of the rearranged groups No 3 for the mod(*Р2*#) represented in Sections 8.1 and 8.3.

That is: 2φ(Р1#) (Section 8.1 with the length = R3 ≤ 2Р2) + φ(Р1#)(Р2–3) (Section 8.3 with the length R3 < < R4 = 2Р2) = Р2φ(Р1#)–3φ(Р1#) + 2φ(Р1#) = *Р<sup>2</sup>* φ(*Р1*#)– φ(*Р1*#) = φ(*Р1*#)(*Р2*–1) = φ(*Р2*#) of the subgroups No 3 for the mod(*Р2*#), included at the alternances ≤*Р<sup>2</sup>* with length R3 ≤ 2*Р*2, including two maximally long subgroups No 3 max R3 = 2*Р*2.

8.5

"eliminates," that is rearranges to the mod(*Р2*#) one time every of Р<sup>2</sup> repeated numbers of the P.I. section from 1 to Р1# (1 number at every Р<sup>2</sup> line of every column No 4 and No 3), "eliminating" the residues С1-С2-С3-С<sup>4</sup> in the groups No 4 (consult Section 8.1), and ALL numbers, besides the residues С1-С2-С<sup>3</sup> in the groups No 3

It is quite obvious, that after "elimination" from every φ(*Р1*#) column of the group No 4 of mod(*Р1*#) one time every residue С<sup>2</sup> and С3, we'll get at *Р<sup>2</sup>* lines of every column of the No 4 groups of mod(*Р1*#), –TWO groups No 3 of mod(*Р2*#) That is, we'll get 2φ(*Р1*#) subgroups No 3 of mod(*Р2*#) with invariance length of the previous groups, that is *R4* φ(*Р1*#) groups of No 4 mod(*Р1*#), would become = *R3* for 2φ(*Р1*#) groups No 3 of mod(*Р2*#), with changing the alternance structure from ≤*Р<sup>1</sup>* to ≤*Р2*, that are situated at the P.I section from 1 to *Р2*# that is at the fractal-*Р2*#. Herewith within the 2φ(*Р1*#) groups No 3 of mod(*Р2*#), there are accounted all residues = С<sup>1</sup> and = С<sup>4</sup> of mod(*Р1*#) and alternances ≤*Р<sup>2</sup>* of such rearranged groups from No 4 of mod(*Р1*#) to No 3 of mod(*Р2*#). As along of *Р<sup>2</sup>* duplication of the three adjoined groups No 4 of mod(*Р1*#), represented in **Table 3**, we'll get **Table 17**, with every residue = С<sup>2</sup> or = С3, situated at one of the 3 lines of **Table 3** (for example, at line а of **Table 17**), is accounted as residue С<sup>1</sup> or С<sup>4</sup> at the other two adjoined groups No 4 (at lines: b or с of **Table 17**), where they are situated within 4 consequent residues, going as the second or the third, that is they are "excluded" as С2 or 3 in

• for line (b) С2, С3. (С<sup>4</sup> = multiple to *Р2*), С<sup>3</sup> we'll get the alternance ≤*Р2*. R*<sup>3</sup>*

• for line (с) С2, (С<sup>1</sup> = multiple to *Р2*), С2, С<sup>3</sup> we'll get the alternance ≤*Р2*. R*<sup>3</sup>*

Thus, "eliminating" that means transition to mod(*Р2*#), one time every 4 residue in φ(*Р1*#) groups No 4 of mod(*Р1*#), at the P.I. sections from 1 to *Р2*#, that is at the fractal -*Р2*#, we'll, get the loopback, represented as 2φ(*Р1*#) groups No 3 of mod (*Р2*#) type: С<sup>1</sup> ррС<sup>2</sup> рр(С2–<sup>3</sup> = multiple *Р2*) ррС3, with the alternances ≤*Р2*, length *R3* ≤ *2Р2*. Including, pure 2 subgroups No 3 of mod(*Р2*#) with length maximal

Herewith, it is quite obvious that every three consequent residues of every subgroup No 3 according to the increasing group of mod(Рn#), represented in **Table 17**, are still within the P.I. section with length not exceeding - 2*Р*n+1 of the whole numbers, as "eliminated" residues С2;3 and С1;4 of mod(Рn#) doesn't change the location of every subgroup No 3. That is, we'll get at the three adjoined groups No 3 lines of **Table 17** for the mod(*Р2*#): type (а) =(СВ–СА) *< 2Р3*; type (b) =

Including pure two subgroups No 3 max R3 = 2*Р*2, located within the maximally long section with length = 2*Р<sup>3</sup>* of the whole numbers. (The rearrangement is studied

Within the *Р<sup>2</sup>* duplications φ(*Р1*#) of the groups No 3 of mod(*Р1*#), number *Р<sup>2</sup>* "eliminated," that is rearranges to the mod(*Р2*#) 1 time every of all previously eliminated numbers of group No 3, except three residues С1–С2–С3. That is, transits to the mod(*Р2*#) (*Р2*–3) of No 3 groups in every φ(*Р1*#) column of No 3 groups. Then at the P.I. section from 1 to *Р2*#, (that is included into fractal -*Р2*#), we'll get the loop back, represented as (*Р2*–3) φ(*Р1*#) of No 3 groups repetition for the

(consult Section 8.3).

*Number Theory and Its Applications*

these adjoined groups (lines):

(С2–С1) *< 2Р3*; type (с) =(С4–С3) *< 23*.

*< 2Р2.*

*< 2Р2.*

R3 = 2*Р*2. 8.2

in Section 6). 8.3

**62**

8.1

As all φ(*Р***2**#) of the subgroups No 3 of mod(*Р2*#) are examined in Sections 8.1 and 8.3, with all eliminated one time residues С<sup>2</sup> or С3, examined in Section 8.1, cannot change the length = R3 ≤ 2*Р2*, none of the 2φ(*Р1*#) groups, rearranged from No 4 to No 3 for the mod(*Р2*#). Herewith the residues С*<sup>1</sup>* and С*<sup>4</sup>* are also accounted in two adjoining groups of **Table 17** as С*<sup>2</sup>* or С*3*. And indiscriminately φ(*Р1*#)(*Р2*–3) groups No 3 for the mod(*Р2*#), examined in Section 8.3 are shorter than limit R3 = 2*Р2.*

So, there are no other ways to make groups No 3 of mod(*Р2*#) with length R3 > 2*Р2,* besides the way to form the maximally long subgroups No 3 with length max R3 = 2*Р2*, represented in Sections 6 and 8.1.

8.6

Thus we got, that for every recurrent = *Р2*, φ(*Р2*#) residues of mod(*Р2*#) situated in the fractal -*Р2*# (*Р<sup>2</sup>* lines of **Table 1**), represented as loop back φ(*Р2*#) of subgroups No 3 (3 residues of mod(*Р2*#), represented in Section 8.4), that is pure THREE consequent prime (A,B,CС) type: *P2 <* (AС-BС*-*CС) < *P3 <sup>2</sup>* at the P.I. section from *Р<sup>2</sup>* to *Р<sup>3</sup> 2* . And further, from *Р<sup>3</sup> <sup>2</sup>* to *Р2*#, pure THREE consequent residue of mod(*Р2*#) at every P.I. section, with length not exceeding 2*Р<sup>3</sup>* of the whole numbers (see Section 8.2), with length of every subgroup No 3 at every section is R3 = (CС–AС) ≤ **2***Р<sup>2</sup>* (consult Sections 8.1 and 8.4).

And so on, for every of all eventual primes **=** *Р*n, represented as the loopback of groups distribution of the residues No 3 at the increasing fractals -*Р*n# according to the increasing meanings of modulus-mod(*Р*n#) that proves the validity of section (b) of the Theorem 1 (loopback of groups No 3 is represented in column No 3 of **Table 14**).

### **9. The loopback of rearrangement for φ(***Р***n#) groups No 2 (2 residues) from the mod(***Р***n-1#) to mod(***Р***n#)**

The looped back order of rearrangement φ(*Р*n#) of No 2 groups, according to the increasing modulus, that are represented in column No 4 of **Table 14**, in Section 9 are examined by steps for every recurrent increasing fractal -*Р*n#:

Representing the first *Р<sup>2</sup>* lines in **Table 1** as one line, we'll get the fractal -*Р2*# according to mod(*Р2*#)-I.R.S of mod(*Р2*#) at the P.I. sections from 1 to *Р2*# (1 line of **Table 2**).

With φ(*Р2*#) line-symmetrical the least residues of mod(*Р2*#), which according to Section 2 are indexed according to φ(*Р2*#) groups of residues No 4-3-2 for the mod(*Р2*#).


At the transition from mod(

**Type-(a) group from 4**

*Prime Numbers Distribution Line*

Type-(b) group from 4

Type-(c) group from 4 to No 3 (

С 2 – С 3 )

P

**to No 3 ( С 1 – С 4 )**

to No 3 ( С 2 – С 3 )

**Table 17.**

*Table 3), while*

*Р2***#)** groups No 4-3-2 mod(

*Р3*#) one time every

*Р2*#. (One number at every

It is quite obvious, that after

*Р2*#), 1 time the residue -

*Р3*# we

**С А С 1 ( С <sup>2</sup> = mtp.** *Р2* **)** *или*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

С 1 С 2 , С 3, ( С

С 3 С 2, ( С

<sup>2</sup> *duplication of fractal -*

**( С <sup>3</sup> = mtp.** *Р2***), С 4**

'll get φ (

*Р2*#), one group No 2 of mod(

*Р2*#) and alternances

С = multiple

*Р2*#) groups No 3 of mod(

<sup>2</sup> = multiple

*Р2*. Including pure 2 subgroups No 2 of mod(

*Р2*#) to No 2 of mod(

а

*Р3*#) with changing the alternance composition from

*Р2*#) groups No 2 of mod(

*Р<sup>3</sup>* according to diagonals

*The representation of rearrangement of every 3 adjoined subgroups No 4 of mod(*

Р

the φ (

1 to

φ (

mod(

and = С

(line):

mod(

*R 2* ≤ *2*

R <sup>2</sup> = 2*Р*2. 9.2

section from 1 to

the mod (

9.1

3 of mod(

No 3 of mod(

subgroups No 2 of mod(

<sup>3</sup> of mod(

(for example, the line (


Thus, after

that is in fractal -

*Р3*#), type:

3 residues at

we get: type (

in Section 7.

**65**

Herewith in

No 3 of mod(

*Р2*#) groups No 3 of mod(

situated at P.I. section from 1 to

φ (

two adjoined groups No 3 of mod(

2 С., ( 3

> С 1 рр ( С

No 2 according to the increasing mod(

Including pure two subgroups No 2 max R

а) =( С В – С А ) *< 2*

φ (

table No 11 with each of the residues =

"elimination

*Р3*#, we

column). Number

*Р2*#) to mod(

<sup>4</sup> = multiple to *Р2*),С<sup>3</sup>

<sup>1</sup> = multiple to *Р2*) С2,С<sup>3</sup>

*Р2*#) are repeated

ing" the residues С1-С2-С<sup>3</sup> at the groups No 3 (consult Section 9.1) and ALL num-

bers, except the residues С1-С<sup>2</sup> at the groups No 2 (consult Section 9.3).

С 2, we

"elimination

*Р2*#) would become =

≤

С

adjoined group No 3 (in line b). of **Table 12**), where it is represented in 3 consequent residues as the second one, that is "excluded" as С<sup>2</sup> at this adjoined group

" that is transferring to mod(

'll get the loop back in the form of

Herewith it is quite obvious that any two consequent residues of any subgroup

С 1 – 1 С ) *< 2 Р3* .

Р

*Р3*; type (b) =(

the P.I. section with length not exceeding - 2*Р*n+1 of the whole numbers, as the "eliminated" residues С<sup>2</sup> and С1;3 of mod(Рn#) doesn't change the location of any subgroup No 2. That is, in two adjoined groups No 2 lines of **Table 18** for mod(

long section with length = 2*Р<sup>3</sup>* of the whole numbers, two rearrangement is studies

*Р3*)., С 2, we

> *Р2* ) рр С

) **Table 18**, is accounted as the residue

*Р3*#), the fractal -

"eliminates,

*Р<sup>3</sup>* of the duplicated numbers of the P.I. section from

φ (

*Р3*#), that is totally we

≤ *Р<sup>2</sup>* to ≤

*Р3*#) are accounted all residues =

2, situated on one of two lines of **Table 5**

С <sup>1</sup> or С

*Р3* #

"rearranged

'll get the alternance

3, with the alternances

*Р2*#), at the P.I. sections from 1 to

φ (

n#), represented in **Table 18** are still within

*Р2*#) columns of No 4-3-2 groups (

*Р<sup>3</sup>* line of every column No 3 and No 2),

" in every

*Р3*#) with invariance length of previous groups, that is

*R <sup>2</sup>* for φ (

*Р2*#), represented in **Table 5**, we

'll get in

*Р3*# that is at the fractal -

*Р<sup>3</sup>* of such

*Р3*#). As in the course of

*Р<sup>3</sup>* lines

**С**

С <sup>2</sup> with R

С <sup>4</sup> with R

<sup>1</sup>*# from No 4 groups of mod(*

**В with R**

С 2 ) < *2 Р2*

С 2 ) < *2 Р2*

**( С 4 – С 1 ) <** *2Р<sup>2</sup>*

**<sup>4</sup> is R 3**

<sup>4</sup> is R 3 ( С 3 –

<sup>4</sup> is R 3 ( С 3 –

Р

*Р2*# and every from

<sup>1</sup>*#) (represented in*

<sup>1</sup>*#) to No 3 groups of mod(*

**at the P.I. section (СВ–СА)** *< 2Р<sup>3</sup>*

at the P.I. section (С2–С1) < *2Р<sup>3</sup>*

at the P.I. section (С<sup>4</sup> – С3) *< 2Р<sup>3</sup>*

*Р<sup>3</sup>* lines at the

"eliminat-

Р <sup>2</sup>*#).*

*R 3*

С 1

<sup>3</sup> at the other

*Р3*#,

*Р3*#)

*Р3*, with length

" that is transits for

*Р2*#) column of group No

'll get φ ( *Р2*#)

*Р2*#) groups No 2 of

" groups from

'll get the

*Р<sup>3</sup>* duplication of

≤ *Р3*. R*<sup>2</sup> < 2 Р2*

≤

*Р3*#) with length maximal

<sup>2</sup> = 2*Р*2, located within the maximally

*Р3*#), one time for every of

*Р2*#) groups No 2 of

*Р3*, that are

*Р<sup>3</sup>* line of every column of groups

*Р<sup>3</sup>* times. Then at the P.I.

Р

**Table** 

*The* 

#### *Number Theory and Its Applications*


**Table 17.**

*The representation of rearrangement of every 3 adjoined subgroups No 4 of mod(*Р1*#) (represented in Table 3), while* P2 *duplication of fractal -*Р1*# from No 4 groups of mod(*Р1*#) to No 3 groups of mod(*Р2*#).*

At the transition from mod(*Р2*#) to mod(*Р3*#), the fractal -*Р2*# and every from the φ(*Р2***#)** groups No 4-3-2 mod(*Р2*#) are repeated *Р<sup>3</sup>* times. Then at the P.I. section from 1 to *Р3*# we'll get φ(*Р2*#) columns of No 4-3-2 groups (*Р<sup>3</sup>* lines at the column). Number *Р<sup>3</sup>* according to diagonals *Р<sup>3</sup>* lines "eliminates," that is transits for the mod (*Р3*#) one time every *Р<sup>3</sup>* of the duplicated numbers of the P.I. section from 1 to *Р2*#. (One number at every *Р<sup>3</sup>* line of every column No 3 and No 2), "eliminating" the residues С1-С2-С<sup>3</sup> at the groups No 3 (consult Section 9.1) and ALL numbers, except the residues С1-С<sup>2</sup> at the groups No 2 (consult Section 9.3). 9.1

It is quite obvious, that after "elimination" in every φ(*Р2*#) column of group No 3 of mod(*Р2*#), 1 time the residue - С2, we'll get in *Р<sup>3</sup>* line of every column of groups No 3 of mod(*Р2*#), one group No 2 of mod(*Р3*#), that is totally we'll get φ(*Р2*#) subgroups No 2 of mod(*Р3*#) with invariance length of previous groups, that is *R3* φ(*Р2*#) groups No 3 of mod(*Р2*#) would become = *R2* for φ(*Р2*#) groups No 2 of mod(*Р3*#) with changing the alternance composition from ≤*Р<sup>2</sup>* to ≤*Р3*, that are situated at P.I. section from 1 to *Р3*# that is at the fractal -*Р3*#

Herewith in φ(*Р2*#) groups No 2 of mod(*Р3*#) are accounted all residues = С<sup>1</sup> and = С<sup>3</sup> of mod(*Р2*#) and alternances ≤*Р<sup>3</sup>* of such "rearranged" groups from No 3 of mod(*Р2*#) to No 2 of mod(*Р3*#). As in the course of *Р<sup>3</sup>* duplication of two adjoined groups No 3 of mod(*Р2*#), represented in **Table 5**, we'll get the table No 11 with each of the residues = С2, situated on one of two lines of **Table 5** (for example, the line (а) **Table 18**, is accounted as the residue С<sup>1</sup> or С<sup>3</sup> at the other adjoined group No 3 (in line b). of **Table 12**), where it is represented in 3 consequent residues as the second one, that is "excluded" as С<sup>2</sup> at this adjoined group (line):


Thus, after "elimination" that is transferring to mod(*Р3*#), one time for every of 3 residues at φ(*Р2*#) groups No 3 of mod(*Р2*#), at the P.I. sections from 1 to *Р3*#, that is in fractal -*Р3*#, we'll get the loop back in the form of φ(*Р2*#) groups No 2 of mod(*Р3*#), type: С<sup>1</sup> рр(С<sup>2</sup> = multiple *Р2*) ррС3, with the alternances ≤*Р3*, with length *R2* ≤ *2Р2*. Including pure 2 subgroups No 2 of mod(*Р3*#) with length maximal R2 = 2*Р*2.

9.2

for the fractal - *7*# = 210, mod(7#), maxR2 = 2\*5, section (

> СА = –1

**64**

…

С1 = 1.

3.

(5) и (7)

3.

С2 = 11

..

.СВ = 13

ВС = 211

2С = 209

С*В*–СА) = 2\*11, n = 4

(7\*29) и (5\*41)

1С = 199

for the fractal - *11*# = 2310, mod(*11*#), maxR2 = 2\*7, section (

АС = 197

.СА = 109.

.

..С1 = 113.

5,

(7\*17),(11\*11)

3,

..С2 = 127 =n5# + 7

… 2С = 2197

(7\*11–n)5# + 7

С*В*–СА) = 2\*13, n = 45

…

…

..

.СВ = 131

*Number Theory and Its Applications*

=n5# + 11

ВС = 2201.

(7\*11–n)5# + 11

5.

3.

=n5#.

 .*<sup>1</sup>* = 120 (11\*191),(7\*313)

=(7\*11–n)5#

 1

 1

=n5#*–7*

1С = 2183.

=11#–С2

for the fractal - *13*# = 30,030, mod(*13*#), maxR2 = 2\*11, section (

…

=n5#*–11*

.АС = 2179.

=11#–СВ

.СА = 9437.

.

..С1 = 9439.

3,

(11\*859),(13\*727)

3,

…

С2 = 9461

..

.СВ = 9463

=n7# + 13 ВС = 20,593..

(11\*13–n)7#

 + 13

=n7# + 11

… 2С = 20,591..

(11\*13-n)7#

 + 11

7,

=n7#.

 .*<sup>1</sup> = 9450* (13\*1583),(11\*1871)

=(11\*13–n)7#

510,510,mod(*17*#),

 maxR2 = 2\*13, section (

С*В*–СА) = 2\*17,n = 94

 1

 *1*

5,

7,

3.

5.

3.

for the fractal - *17*# =

=n7#*–11*

1С = 20,569.

=13#–С2

…

=n7#*–13*

.АС = 20,567.

=13#–СВ

.СА = 217,123.

.

..С1 = 217,127.

11,

(13\*16703),(17\*12773)

3

..С2 = 217,153..

..

.СВ = 217,157

=n11# + 17 ВС = 293,387

(13\*17–n)11#

 + 17

=n11# + 13 (13\*17–n)11#

 + 13

3,

=n11#.

 .*<sup>1</sup> = 217,140* (17\*17257),(13\*22567)

=(13\*17–n)11#

 1

 *1*

5,

7,

… 2С = 293,383

3,

11,

7,

5,

3.

for the fractal - *19*# =

9,699,690,mod(*19*#),

 maxR2 = 2\*17, section (

С*В –*СА) = 2\*19, n = 2

=n11#*–13*

1С = 293,357.

=17#–С2

…

=n11#*–17*

.АС = 293,353.

=17#–СВ

СА = 60,041

…

С1 = 60,043

13

кр.19., кр.17 = 60,061

3,

..С2 = 60,077

..

 .

ВС = 9,639,649

СВ = 60,079

> 2С = 9,639,647

.13.

Кр.17.,кр.19

 = 9,639,631

3

1С = 9,639,613

АС = 9,639,611

**Table 16.** *The numerical*

 *examples of the two* 

*line-symmetrical*

 *maximally*

 *long subgroups No 2 (containing*

 *2 residues), according to the increasing modulus.*

С*В*–СА) = 2\*7, n = 1

> Herewith it is quite obvious that any two consequent residues of any subgroup No 2 according to the increasing mod(Рn#), represented in **Table 18** are still within the P.I. section with length not exceeding - 2*Р*n+1 of the whole numbers, as the "eliminated" residues С<sup>2</sup> and С1;3 of mod(Рn#) doesn't change the location of any subgroup No 2. That is, in two adjoined groups No 2 lines of **Table 18** for mod(*Р3*#) we get: type (а) =(СВ–СА) *< 2Р3*; type (b) =(С1–1С) *< 2Р3*.

> Including pure two subgroups No 2 max R2 = 2*Р*2, located within the maximally long section with length = 2*Р<sup>3</sup>* of the whole numbers, two rearrangement is studies in Section 7.

9.3

Along with *Р<sup>3</sup>* duplications φ(*Р2*#) of groups No 2 of mod(*Р2*#), the number *Р<sup>3</sup>* "eliminates" that is transits to the mod(*Р3*#) 1 time every of all previously eliminated numbers of every group No 2, except two residues С1–С2. That it, it transits to mod(*Р3*#) (*Р3*–2) of groups No 2 in every φ(*Р2*#) column of No 2 groups.

**10. Proof of section (b) and section (с) of Theorem 1**

amounts of different first primes ≤*Р*<sup>n</sup> – NOT residues mod(*Рn*#).

period = *Р*n#).

of mod(*Р***3**#).

(*Р*(1)#)).

axiom set.

**67**

groups No 4-3-2 of mod(*Р***3**#).

*Prime Numbers Distribution Line*

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

residues) of mod(*Р(1)*#) with length maxR4 = 2*Р*(2).

While examining the P.I., represented in the form of alternance (array) of primes (according to the 1 least prime factor > 1 from every whole number), we'll get that for every recurrent prime -*Рn*; the P.I. is the line-symmetrical primaryrepeated fractal -*Р*n#, located at the P.I. section from 1 to *Рn*#, represented as φ(*Рn*#) of the I.R.S. residue of mod(*Рn*#), between which the P.I. sections are situated (with different length), represented as the alternances (array) of different

By indexing φ(*Рn*#) of the least residues of every recurrent fractal -*Рn*#, groups pure with 4, 3, and 2 consequent residue of mod(*Рn*#) (analogously as in Section 2). We'll get that every recurrent fractal -*Рn*# has got three types of arrays of the subgroups of residues of mod(*Рn*#): with φ(*Рn*#) groups: No 4 (with 4 residues), No 3 (with 3 residues), and No 2 (with 2 residues, repeated without changes with the

At every recurrent transition, for example, from mod(*Р***2**#) to mod(*Р***3**#), the first line of fractal -*Р2*# and group No 4-3-2 of mod(*Р***2**#) are repeated *Р<sup>3</sup>* times (consult *Р<sup>3</sup>* lines of **Table 2**). At the P.I. section from 1 to *Р3*# we'll get φ(*Р2*#) columns of groups No 4-3-2 of mod(*Р2*#) within the alternances ≤Р<sup>2</sup> (*Р<sup>3</sup>* lines in column), "eliminating" 1 number multiple to -*Р<sup>3</sup>* (in line of every column No 4-3-2), that is by transition of this group to mod(*Р3*#) with changing of its length: from R(4,3) to R3,2 and the alternance composition from ≤Р<sup>2</sup> to ≤Р3, we'll get φ(*Р3*#)

The rearrangement order of these subgroups No 4-3-2 at the increasing modulus is proved in Sections 5, 6, 7, 8, and 9. Representing as one line of the P.I. section from 1 to *Р3*# we'll get the fractal -*Р***3#** of mod(*Р***3**#), with φ(*Р3*#) groups No 4-3-2

And so on for every recurrent prime = *Рn*, the results of proof are demonstrated in Sections from 5 to 9 and for visualization they are grouped together in **Table 14**. As far as we know that for every recurrent prime -*Р<sup>n</sup>* = *Р(1)*, the P.I. is the linesymmetrical primordial repeated fractal -*Р(1)*#, located at the P.I. section from 1 to *Р(1)*#. There are φ(*Р***(1)#**) residues of mod(*Р***(1)#**) between which are located the alternances (arrays) of primes ≤ *Р*(1) (1 the least>1 from every NOT residue of mod

According to Section 5, there is the only maximally long subgroup No 4 (with 4

Let us assume that there are such primes *Р*(2) or *Р*(3), for which at the P.I., represented as alternance ≤ *Р*(2), in the form of fractal –*Р(2)*# we can form more than two maximally long subgroups No 3 of mod(*Р(2)*#), with: (or) R3 > 2*Р*(2) or at the fractal –*Р*(3)# with P.I. is represented by alternances ≤ *Р*(3), we can compare more than maximally long subgroups No 2 of mod(*Р(3)*#), with: (or) R2 > 2*Р*(2). Then, in the course of the opposite reduction of the modulus, that is at the result of *Р*(2)\**Р*(3) repetition of such subgroups as No 3 or No 2 with the repetition period *Р*(1)# (for the downward meanings of numbers) and backing up the *Р*(2) and *Р(3)* numbers as residues (according to the decreased moduli)**\*\***, that are situated in *Р*(2)\**Р*(3) lines analogues to **Table 1**. At the upper lines of such columns, consisting of *Р*(2)\**Р*(3) lines, we'll get the P.I as fractal-*Р*(1)#, represented by alternances ≤ *Р*(1), where at the P.I. section from 1 to *Р(1)*# would be located more than one subgroup No 4 (with 4 residues) of mod(*Р*(1)#) or subgroups No 4, with R4 > 2*Р*(2). It is quite obvious, that any such group No 4 according to the reestablished mod(*Р*(1)#) would be line-symmetrical to the left and to the right from the symmetry center of number = *Р(1)*#/2, that means formed by two different ways, that contradicts to the

Then at the P.I. section from 1 to *Р3*# that is within the fractal -*Р3*# we'll get the loopback, represented as (*Р3*–2)φ(*Р2*#) duplications of No 2 groups for the mod (*Р3*#) with the alternance "changes" from ≤*Р<sup>2</sup>* to ≤*Р3*. Without changing the length of groups No 2 for mod(*Р3*#), *R2* = const and numbers composition at the alternances ≤*Р<sup>3</sup>* (as previously eliminated ≤*Р2*, for the 1 least >1 from the number is accounted). Type: С<sup>1</sup> ррррС2. With: *R2* = const =? (with to mod(*Р3*#) R2 *< <* R3 = 2*Р*2).

9.4

Totally at the fractal Р3# at P.I. section from 1 to Р3# we'll get the loopback of the rearranged groups No 2 for mod(Р3#) represented in Sections 9.1 and 9.3. That is: φ(Р2#) (Section 9.1 with length = R2 ≤ 2Р2) + φ(Р2#)(Р3–2) (Section 9.3 with length R2 < <R3 = 2Р2) = Р3φ(Р2#)–2φ(Р2#) + φ(Р2#) = Р<sup>3</sup> φ(Р2#)–φ(Р2#) = φ(Р2#)(Р3– 1) = φ(Р3#) of the subgroups No 2 for mod(Р3#), located within the alternances ≤Р<sup>3</sup> with length R2 ≤ 2Р2, including two maximally long subgroups No 2 max R2 = 2Р2. 9.5

As all φ(*Р***3**#) of the subgroups No 2 of mod(*Р3*#) are examines in Sections 9.1 and 9.3, with every eliminated one time in the column residue С<sup>2</sup> examined in Section 9.1, can change the length = R2 ≤ 2*Р2*, of none of φ(*Р2*#) groups, rearranged from No 3 to No 2 for mod(*Р3*#), herewith residues С*<sup>1</sup>* and С*<sup>3</sup>* are excluded at the adjoined group of **Table 18** as С*<sup>2</sup>* and indiscriminately (*Р3*–2)φ(*Р2*#) groups No 2 of mod(*Р2*#), examined in Section 9.3 are shorter than limit R = 2*Р2.*

Thus, there are no other ways of making groups No 2 of mod(*Р3*#) with length R2 > 2*Р2*, but constructing two maximally long subgroups No 2 with length max R2 = 2*Р2*, as examined in Sections 7 and 9.1. 9.6

And so we get, that for every recurrent prime = *Р3*, φ(*Р3*#) residues of mod (*Р3*#) are in the fractal -*Р3*# (in *Р<sup>3</sup>* lines of **Table 2**), represented as loopback φ(*Р3*#) of the subgroups No 2 (2 residues of mod(*Р3*#) are indicated in Section 9.4). Thus, pure TWO consequent primes (A,BС) of type: *P1*, *P2 <* (AС-BС) < *P3 <sup>2</sup>* at the P.I. section from *Р<sup>2</sup>* to *Р<sup>3</sup> 2* , and further, from *Р<sup>3</sup> <sup>2</sup>* to *Р3*# pure TWO consequent residues of mod (*Р3*#), at every P.I. sections with length not exceeding 2*Р<sup>3</sup>* of the whole numbers (consult Section 9.2), with length of every subgroup No 2 at every section is: R2 = (ВС–AС) ≤ **2***Р<sup>2</sup>* (consult Sections 9.1 and 9.4).

And so on, for every from all eventual primes **=** *Р*n, in the form of loopback of residues of groups No 2 distribution at the increasing fractals -*Р*n#, with the increasing values of modulus of mod(*Р*n#), that proofs the validity of section (с) of Theorem 1 (loopback of groups No 2 is represented in column No 4 of **Table 14**).


#### **Table 18.**

*Representation of rearrangement of any 2 adjoined subgroups No 3 of mod(*Р2*#) (represented in Table 5), within* P3 *duplications of fractal -*Р2*#. From groups No 3 of mod (*Р2*#) to the groups No 2 of mod (*Р3*#).*

9.3

*Number Theory and Its Applications*

R3 = 2*Р*2). 9.4

9.5

9.6

section from *Р<sup>2</sup>* to *Р<sup>3</sup>*

**Type-(a) group from 3 to**

Type (a) group from 3 to

**No 2 (С1–С3)**

No 2 (2С–С2)

**Table 18.**

**66**

Along with *Р<sup>3</sup>* duplications φ(*Р2*#) of groups No 2 of mod(*Р2*#), the number *Р<sup>3</sup>* "eliminates" that is transits to the mod(*Р3*#) 1 time every of all previously eliminated numbers of every group No 2, except two residues С1–С2. That it, it transits to

Then at the P.I. section from 1 to *Р3*# that is within the fractal -*Р3*# we'll get the loopback, represented as (*Р3*–2)φ(*Р2*#) duplications of No 2 groups for the mod (*Р3*#) with the alternance "changes" from ≤*Р<sup>2</sup>* to ≤*Р3*. Without changing the length

alternances ≤*Р<sup>3</sup>* (as previously eliminated ≤*Р2*, for the 1 least >1 from the number is

Totally at the fractal Р3# at P.I. section from 1 to Р3# we'll get the loopback of the rearranged groups No 2 for mod(Р3#) represented in Sections 9.1 and 9.3. That is: φ(Р2#) (Section 9.1 with length = R2 ≤ 2Р2) + φ(Р2#)(Р3–2) (Section 9.3 with length R2 < <R3 = 2Р2) = Р3φ(Р2#)–2φ(Р2#) + φ(Р2#) = Р<sup>3</sup> φ(Р2#)–φ(Р2#) = φ(Р2#)(Р3– 1) = φ(Р3#) of the subgroups No 2 for mod(Р3#), located within the alternances ≤Р<sup>3</sup> with length R2 ≤ 2Р2, including two maximally long subgroups No 2 max R2 = 2Р2.

As all φ(*Р***3**#) of the subgroups No 2 of mod(*Р3*#) are examines in Sections 9.1 and 9.3, with every eliminated one time in the column residue С<sup>2</sup> examined in Section 9.1, can change the length = R2 ≤ 2*Р2*, of none of φ(*Р2*#) groups, rearranged from No 3 to No 2 for mod(*Р3*#), herewith residues С*<sup>1</sup>* and С*<sup>3</sup>* are excluded at the adjoined group of **Table 18** as С*<sup>2</sup>* and indiscriminately (*Р3*–2)φ(*Р2*#) groups No 2 of

Thus, there are no other ways of making groups No 2 of mod(*Р3*#) with length R2 > 2*Р2*, but constructing two maximally long subgroups No 2 with length max

And so we get, that for every recurrent prime = *Р3*, φ(*Р3*#) residues of mod (*Р3*#) are in the fractal -*Р3*# (in *Р<sup>3</sup>* lines of **Table 2**), represented as loopback φ(*Р3*#) of the subgroups No 2 (2 residues of mod(*Р3*#) are indicated in Section 9.4). Thus,

of mod (*Р3*#), at every P.I. sections with length not exceeding 2*Р<sup>3</sup>* of the whole numbers (consult Section 9.2), with length of every subgroup No 2 at every section

residues of groups No 2 distribution at the increasing fractals -*Р*n#, with the

**СА С1, (С<sup>2</sup> = multiple** *Р3***)., С<sup>3</sup>**

> (3С<sup>1</sup> = multiple *Р3*).,С<sup>2</sup>

<sup>1</sup>С <sup>2</sup>С.,

And so on, for every from all eventual primes **=** *Р*n, in the form of loopback of

increasing values of modulus of mod(*Р*n#), that proofs the validity of section (с) of Theorem 1 (loopback of groups No 2 is represented in column No 4 of **Table 14**).

*Representation of rearrangement of any 2 adjoined subgroups No 3 of mod(*Р2*#) (represented in Table 5), within* P3 *duplications of fractal -*Р2*#. From groups No 3 of mod (*Р2*#) to the groups No 2 of mod (*Р3*#).*

**СВ with R3 is R2 (С3–С1) <** *2Р<sup>2</sup>*

С<sup>1</sup> with R3 is R2 (С2–2С) *< 2Р<sup>2</sup>* *<sup>2</sup>* at the P.I.

**at the P.I. section (СВ–СА)** *< 2Р<sup>3</sup>*

At the P.I. section (С1–1С) *< 2Р<sup>3</sup>*

*<sup>2</sup>* to *Р3*# pure TWO consequent residues

mod(*Р2*#), examined in Section 9.3 are shorter than limit R = 2*Р2.*

pure TWO consequent primes (A,BС) of type: *P1*, *P2 <* (AС-BС) < *P3*

, and further, from *Р<sup>3</sup>*

R2 = 2*Р2*, as examined in Sections 7 and 9.1.

*2*

is: R2 = (ВС–AС) ≤ **2***Р<sup>2</sup>* (consult Sections 9.1 and 9.4).

mod(*Р3*#) (*Р3*–2) of groups No 2 in every φ(*Р2*#) column of No 2 groups.

of groups No 2 for mod(*Р3*#), *R2* = const and numbers composition at the

accounted). Type: С<sup>1</sup> ррррС2. With: *R2* = const =? (with to mod(*Р3*#) R2 *< <*

#### **10. Proof of section (b) and section (с) of Theorem 1**

While examining the P.I., represented in the form of alternance (array) of primes (according to the 1 least prime factor > 1 from every whole number), we'll get that for every recurrent prime -*Рn*; the P.I. is the line-symmetrical primaryrepeated fractal -*Р*n#, located at the P.I. section from 1 to *Рn*#, represented as φ(*Рn*#) of the I.R.S. residue of mod(*Рn*#), between which the P.I. sections are situated (with different length), represented as the alternances (array) of different amounts of different first primes ≤*Р*<sup>n</sup> – NOT residues mod(*Рn*#).

By indexing φ(*Рn*#) of the least residues of every recurrent fractal -*Рn*#, groups pure with 4, 3, and 2 consequent residue of mod(*Рn*#) (analogously as in Section 2). We'll get that every recurrent fractal -*Рn*# has got three types of arrays of the subgroups of residues of mod(*Рn*#): with φ(*Рn*#) groups: No 4 (with 4 residues), No 3 (with 3 residues), and No 2 (with 2 residues, repeated without changes with the period = *Р*n#).

At every recurrent transition, for example, from mod(*Р***2**#) to mod(*Р***3**#), the first line of fractal -*Р2*# and group No 4-3-2 of mod(*Р***2**#) are repeated *Р<sup>3</sup>* times (consult *Р<sup>3</sup>* lines of **Table 2**). At the P.I. section from 1 to *Р3*# we'll get φ(*Р2*#) columns of groups No 4-3-2 of mod(*Р2*#) within the alternances ≤Р<sup>2</sup> (*Р<sup>3</sup>* lines in column), "eliminating" 1 number multiple to -*Р<sup>3</sup>* (in line of every column No 4-3-2), that is by transition of this group to mod(*Р3*#) with changing of its length: from R(4,3) to R3,2 and the alternance composition from ≤Р<sup>2</sup> to ≤Р3, we'll get φ(*Р3*#) groups No 4-3-2 of mod(*Р***3**#).

The rearrangement order of these subgroups No 4-3-2 at the increasing modulus is proved in Sections 5, 6, 7, 8, and 9. Representing as one line of the P.I. section from 1 to *Р3*# we'll get the fractal -*Р***3#** of mod(*Р***3**#), with φ(*Р3*#) groups No 4-3-2 of mod(*Р***3**#).

And so on for every recurrent prime = *Рn*, the results of proof are demonstrated in Sections from 5 to 9 and for visualization they are grouped together in **Table 14**.

As far as we know that for every recurrent prime -*Р<sup>n</sup>* = *Р(1)*, the P.I. is the linesymmetrical primordial repeated fractal -*Р(1)*#, located at the P.I. section from 1 to *Р(1)*#. There are φ(*Р***(1)#**) residues of mod(*Р***(1)#**) between which are located the alternances (arrays) of primes ≤ *Р*(1) (1 the least>1 from every NOT residue of mod (*Р*(1)#)).

According to Section 5, there is the only maximally long subgroup No 4 (with 4 residues) of mod(*Р(1)*#) with length maxR4 = 2*Р*(2).

Let us assume that there are such primes *Р*(2) or *Р*(3), for which at the P.I., represented as alternance ≤ *Р*(2), in the form of fractal –*Р(2)*# we can form more than two maximally long subgroups No 3 of mod(*Р(2)*#), with: (or) R3 > 2*Р*(2) or at the fractal –*Р*(3)# with P.I. is represented by alternances ≤ *Р*(3), we can compare more than maximally long subgroups No 2 of mod(*Р(3)*#), with: (or) R2 > 2*Р*(2).

Then, in the course of the opposite reduction of the modulus, that is at the result of *Р*(2)\**Р*(3) repetition of such subgroups as No 3 or No 2 with the repetition period *Р*(1)# (for the downward meanings of numbers) and backing up the *Р*(2) and *Р(3)* numbers as residues (according to the decreased moduli)**\*\***, that are situated in *Р*(2)\**Р*(3) lines analogues to **Table 1**. At the upper lines of such columns, consisting of *Р*(2)\**Р*(3) lines, we'll get the P.I as fractal-*Р*(1)#, represented by alternances ≤ *Р*(1), where at the P.I. section from 1 to *Р(1)*# would be located more than one subgroup No 4 (with 4 residues) of mod(*Р*(1)#) or subgroups No 4, with R4 > 2*Р*(2). It is quite obvious, that any such group No 4 according to the reestablished mod(*Р*(1)#) would be line-symmetrical to the left and to the right from the symmetry center of number = *Р(1)*#/2, that means formed by two different ways, that contradicts to the axiom set.

\*\*Number *Р*(3) of the fractal -*Р*(3)# -NOT residue of mod(*Р3*#), located in the group No 2 of mod(*Р(3)*#) within the alternance ≤ *Р*(3) with length R(2) > 2*Р*(2), and for mod(*Р(2)*#) would be accounted as the third residue in the group No 3 within the alternance ≤*Р*(2), without changing the length of this group No 3 with R(2) is R3 > 2*Р*(2).

whole numbers, there is the subgroup from two consequent primes *Р*<sup>1</sup> and *Р*2.

<sup>p</sup> whole numbers <sup>&</sup>gt;2*Р*<sup>1</sup> whole numbers, that means, that at the P.I.

whole numbers, there is the loopback of primes, represented as the subgroup

*<sup>2</sup>* as, *Р*<sup>1</sup>

where there are no two primes, that is, two consequent primes are located at the P.I.

dicts to section (с) of the Theorem "Loopback of primes distribution," that states, that is every fractal -*Р1*# according to mod(*Р1*#), on every P.I. sections with length not exceeding **2***Р<sup>1</sup>* of the whole numbers, there is a subgroup No 2 with 2 residues of

<sup>p</sup> at every P.I. section with length not exceeding 2 ffiffiffiffi

<sup>2</sup> < (N + 2 ffiffiffiffi

<sup>p</sup> of the whole numbers is located

2 , with:

<sup>p</sup> of the whole numbers,

*N* <sup>p</sup> <sup>Þ</sup> <sup>≤</sup> *<sup>Р</sup>*<sup>2</sup>

*N*

<sup>p</sup> of the whole numbers <sup>&</sup>gt;2*Р1*, but this contra-

*N*

*N* p

As 2 ffiffiffiffi *N*

2 ffiffiffiffi *N* <sup>p</sup> <sup>&</sup>gt; <sup>2</sup>*Р*1.

section from 1 to N + 2 ffiffiffiffi

*DOI: http://dx.doi.org/10.5772/intechopen.92639*

*Prime Numbers Distribution Line*

for two consequent primes.

at the fractal -*Р1*# at the P.I. section. <*Р<sup>2</sup>*

section with length exceeding – 2 ffiffiffiffi

mod(*Р1*#), that is two primes <*Р<sup>2</sup>*

**Author details**

**69**

Shcherbakov Aleksandr Gennadiyevich Independent Researcher, Russia

provided the original work is properly cited.

\*Address all correspondence to: ag\_ask@mail.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*N*

It is feasible, that there is a P.I. section with length = 2 ffiffiffiffi

*N*

*2* .

Genuinely: Every P.I. section with length = 2 ffiffiffiffi

Number *Р*(2) of the fractal -*Р*(2)# -NOT residue of mod(*Р2*#) is located in the group No 3 of mod(*Р(2)*#) within the alternance ≤*Р*(2), with length R3 > 2*Р*(2), and for mod

(*Р(1)*#), would be accounted as the fourth residue in the group No 4 within the alternance ≤ *Р*(1), without changing the length of this group No 4 with R(3) is R4 > 2*Р*(2).

#### **11. Conclusion**

Thus, Theorem 1 allowed us to prove the existence of a new law in mathematics – "on the plume distribution of Prime numbers." Since the methods used in number theory do not allow us to approach the problem of the distribution of prime numbers, it means that further expansion of the method proposed in the article for studying the natural series of numbers will simplify and solve many other problems that are not solved in mathematics.

So from Theorem 1 "Loopback of primes distribution" follows:

**Theorem No 2.** For every whole number = N at the P.I. section from 1 to N + 2 ffiffiffiffi *N* <sup>p</sup> :


Proof. Every whole number = N is located within the squared two consequent primes: *Р*<sup>1</sup> <sup>2</sup> < N ≤ *Р*<sup>2</sup> <sup>2</sup> with: 2 ffiffiffiffi *N* <sup>p</sup> <sup>&</sup>gt;2*Р*1.

That means every N is located within the fractal -*Р*1#. Then:

1.From the section (b) of the theorem "Loopback of primes distribution" follows, that at fractal -*Р1*# of mod(*Р1*#) at the P.I. section from 1 to *Р*<sup>2</sup> <sup>2</sup> ≥ (N + 2 ffiffiffiffi *N* <sup>p</sup> ), at every P.I. section with length not exceeding 2*Р<sup>2</sup>* of the whole numbers is located at the subgroup from three consequent primes of (*Р*1-*Р*2- *Р*3) type, with length of every subgroup, that is distance from the first to the third prime of every subgroup doesn't exceed 2*Р*<sup>1</sup> whole numbers, that is (*Р*3–*Р1*) < **2***Р<sup>1</sup>* whole numbers. As length of every section 2 ffiffiffiffi *N* <sup>p</sup> <sup>&</sup>gt; length of the section = 2*Р1*. Then from 1 to N + 2 ffiffiffiffi *N* <sup>p</sup> – every:

(*Р*3–*Р1*) < 2 ffiffiffiffi *N* <sup>p</sup> of the whole numbers.

2.From the section (с) of the Theorem "Loopback of primes distribution" it follows, that at the fractal -*Р1*# of mod(*Р1*#) at the P.I. section from 1 to *Р*2 <sup>2</sup> ≥ (N + 2 ffiffiffiffi *N* <sup>p</sup> ) at every I.P. section with length not exceeding 2*Р<sup>1</sup>* of the

\*\*Number *Р*(3) of the fractal -*Р*(3)# -NOT residue of mod(*Р3*#), located in the group No 2 of mod(*Р(3)*#) within the alternance ≤ *Р*(3) with length R(2) > 2*Р*(2), and for mod(*Р(2)*#) would be accounted as the third residue in the group No 3 within the alternance ≤*Р*(2), without changing the length of this group No 3 with R(2) is

Number *Р*(2) of the fractal -*Р*(2)# -NOT residue of mod(*Р2*#) is located in the group No 3 of mod(*Р(2)*#) within the alternance ≤*Р*(2), with length R3 > 2*Р*(2), and

(*Р(1)*#), would be accounted as the fourth residue in the group No 4 within the

alternance ≤ *Р*(1), without changing the length of this group No 4 with R(3) is

Thus, Theorem 1 allowed us to prove the existence of a new law in

So from Theorem 1 "Loopback of primes distribution" follows:

problems that are not solved in mathematics.

*N*

at every P.I. sections, shorter than 2 ffiffiffiffi

section = 2*Р1*. Then from 1 to N + 2 ffiffiffiffi

<sup>p</sup> of the whole numbers.

<sup>2</sup> with: 2 ffiffiffiffi

*N* <sup>p</sup> <sup>&</sup>gt;2*Р*1. That means every N is located within the fractal -*Р*1#. Then:

group is less than 2 ffiffiffiffi

<sup>2</sup> < N ≤ *Р*<sup>2</sup>

numbers.

primes: *Р*<sup>1</sup>

2 ffiffiffiffi *N*

(*Р*3–*Р1*) < 2 ffiffiffiffi

<sup>2</sup> ≥ (N + 2 ffiffiffiffi

*Р*2

**68**

*N*

*N*

mathematics – "on the plume distribution of Prime numbers." Since the methods used in number theory do not allow us to approach the problem of the distribution of prime numbers, it means that further expansion of the method proposed in the article for studying the natural series of numbers will simplify and solve many other

**Theorem No 2.** For every whole number = N at the P.I. section from 1 to N +

1.Primes are located as groups, pure three consequent primes of (*Р*1-*Р*2-*Р*3) type. Herewith the distance from the first to the third prime of every

2.The same primes are distributed as the loopback, pure two consequent primes

*N*

Proof. Every whole number = N is located within the squared two consequent

1.From the section (b) of the theorem "Loopback of primes distribution" follows, that at fractal -*Р1*# of mod(*Р1*#) at the P.I. section from 1 to *Р*<sup>2</sup>

(*Р*3–*Р1*) < **2***Р<sup>1</sup>* whole numbers. As length of every section 2 ffiffiffiffi

<sup>p</sup> ), at every P.I. section with length not exceeding 2*Р<sup>2</sup>* of the whole numbers is located at the subgroup from three consequent primes of (*Р*1-*Р*2- *Р*3) type, with length of every subgroup, that is distance from the first to the third prime of every subgroup doesn't exceed 2*Р*<sup>1</sup> whole numbers, that is

> *N* <sup>p</sup> – every:

2.From the section (с) of the Theorem "Loopback of primes distribution" it follows, that at the fractal -*Р1*# of mod(*Р1*#) at the P.I. section from 1 to

<sup>p</sup> ) at every I.P. section with length not exceeding 2*Р<sup>1</sup>* of the

<sup>p</sup> of the whole numbers, that is (*Р*3–*Р*1) <sup>&</sup>lt; <sup>2</sup> ffiffiffiffi

<sup>p</sup> whole numbers.

*N* <sup>p</sup> whole

<sup>2</sup> ≥ (N +

*N*

<sup>p</sup> <sup>&</sup>gt; length of the

R3 > 2*Р*(2).

*Number Theory and Its Applications*

for mod

R4 > 2*Р*(2).

2 ffiffiffiffi *N* <sup>p</sup> :

**11. Conclusion**

whole numbers, there is the subgroup from two consequent primes *Р*<sup>1</sup> and *Р*2. As 2 ffiffiffiffi *N* <sup>p</sup> whole numbers <sup>&</sup>gt;2*Р*<sup>1</sup> whole numbers, that means, that at the P.I. section from 1 to N + 2 ffiffiffiffi *N* <sup>p</sup> at every P.I. section with length not exceeding 2 ffiffiffiffi *N* p whole numbers, there is the loopback of primes, represented as the subgroup for two consequent primes.

Genuinely: Every P.I. section with length = 2 ffiffiffiffi *N* <sup>p</sup> of the whole numbers is located at the fractal -*Р1*# at the P.I. section. <*Р<sup>2</sup> <sup>2</sup>* as, *Р*<sup>1</sup> <sup>2</sup> < (N + 2 ffiffiffiffi *N* <sup>p</sup> <sup>Þ</sup> <sup>≤</sup> *<sup>Р</sup>*<sup>2</sup> 2 , with: 2 ffiffiffiffi *N* <sup>p</sup> <sup>&</sup>gt; <sup>2</sup>*Р*1.

It is feasible, that there is a P.I. section with length = 2 ffiffiffiffi *N* <sup>p</sup> of the whole numbers, where there are no two primes, that is, two consequent primes are located at the P.I. section with length exceeding – 2 ffiffiffiffi *N* <sup>p</sup> of the whole numbers <sup>&</sup>gt;2*Р1*, but this contradicts to section (с) of the Theorem "Loopback of primes distribution," that states, that is every fractal -*Р1*# according to mod(*Р1*#), on every P.I. sections with length not exceeding **2***Р<sup>1</sup>* of the whole numbers, there is a subgroup No 2 with 2 residues of mod(*Р1*#), that is two primes <*Р<sup>2</sup> 2* .

#### **Author details**

Shcherbakov Aleksandr Gennadiyevich Independent Researcher, Russia

\*Address all correspondence to: ag\_ask@mail.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Bergman GN. Chislo i nauka o nem. [A Number and A Theory of Numbers]. Moscow: Gostekhizdat; 1949. [in Russian]

[2] Vinogradov IM. Osnovy teorii chisel. [Fundamentals of A Theory of Numbers]. Moscow: Gostekhizdat; 1952. [in Russian]

[3] Ozhigova YP. Razvitie teorii chisel v Rossii. [Development of A Theory of Numbers in Russia]. 2nd ed. Moscow: Directory of the Ukrainian SSR; Vol. 38. 2003. pp. 270-275

[4] Serpinsky V. Chto my znaem i chego ne znaem o prostyh chislah. [What Do We Know and What We Don't Know About Simple Numbers]. Moscow: State Publishing House for physical and mathematical Literature; 1963. [in Russian]

[5] Aleksandrov PS. Jenciklopedija jelementarnoj matematiki/ kniga pervaja arifmetika// [Encyclopedia of Elementary Mathematics. Volume 1 Arithmetic]. Izdatel'stvo tehnikoteoreticheskoj literatury; Moscow. 1951. pp. 255-282. [in Russian]

[6] Prahar K. Raspredelenie prostyh chisel. Izdatel'stvo"MIR"; Moscow. 1967. p. 511

[7] Shherbakov AG. O raspredelenii prostyh chisel. [On Distribution of Prime Numbers]. Vol. 10. Perspektivy nauki. - Tambov, Izdatel'stvo MOO Fond razvitija nauki i kul'tury; 2013. pp. S142-S147. [in Russian]

[8] Shcherbakov AG. Chetyre prostykh chisla mezhdu kvadratami dvukh posledovatelnykh prostykh chisel. In: NAUKA SEGODNYa globalnyye vyzovy i mekhanizmy razvitiya: materialy Mezhdunarodnoy nauchno-prakticheskoy konferentsii. Vologda: Disput; 2018. pp. 77-84

[9] Shherbakov AG. Dokazatel'stvo Gipotezy Brokara (O chetyreh prostyh chislah). In: Aspirant. Vol. 12. Prilozhenie k zhurnalu Vestnik ZBGU; 2018. pp. S.109-S.116

**Chapter 5**

**Abstract**

P*<sup>n</sup>*þ<sup>1</sup>

section.

moments

**1. Introduction**

combinatorics.

**71**

examples and others in [19, 20].

*<sup>k</sup>*¼<sup>1</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup>*

Numbers

Moments of Catalan Triangle

In this chapter, we consider the Catalan numbers, *Cn* <sup>¼</sup> <sup>1</sup>

these Catalan triangle numbers, i.e., with the following sums: P*<sup>n</sup>*

their generalizations, Catalan triangle numbers, *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*, for *n*, *k*∈ . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of

sions for some values of *m* and *j*. Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last

**Keywords:** Catalan numbers, combinatorial identities, binomial coefficients,

After the binomial coefficients, the well-known Catalan numbers ð Þ *Cn <sup>n</sup>* <sup>≥</sup><sup>0</sup> are the most frequently occurring combinatorial numbers. They are treated deeply in many books, monographs, and papers (e.g., [1–20]). Catalan numbers play an

They appear in studying astonishingly many combinatorial problems. They count the number of different ways to triangulate a regular polygon with *n* þ 2 sides; or, the number of ways that 2*n* people seat around a circular table are simultaneously shaking hands with another person at the table in such a way that none of the arms cross each other, and also in tree enumeration problem, see these

Other applications of the Catalan numbers appear in engineering in the field of cryptography to form keys for secure transfer of information; in computational geometry, they are generally used in geometric modeling; they may be also found in

important role and have a major importance in computer science and

geographic information systems, geodesy, or medicine.

*<sup>n</sup>*,*<sup>k</sup>*, for *j*, *n* ∈ and *m* ∈ ∪ f g0 . We present their closed expres-

*n*þ1

2*n n*

*<sup>k</sup>*¼<sup>1</sup>*kmBj <sup>n</sup>*,*<sup>k</sup>*,

� �, and two of

*Pedro J. Miana and Natalia Romero*

[10] Shherbakov AG. The Length Of An Interval Of A Positive Integers Sequence Represented (Measured) By An Alternation Of The First Prime Numbers. GJPAM. 2015;**11**(4): 1803-1818

#### **Chapter 5**

**References**

Russian]

[in Russian]

2003. pp. 270-275

[in Russian]

1967. p. 511

pp. 77-84

**70**

[1] Bergman GN. Chislo i nauka o nem. [A Number and A Theory of Numbers]. Moscow: Gostekhizdat; 1949. [in

[9] Shherbakov AG. Dokazatel'stvo Gipotezy Brokara (O chetyreh prostyh

Prilozhenie k zhurnalu Vestnik ZBGU;

[10] Shherbakov AG. The Length Of An Interval Of A Positive Integers Sequence

chislah). In: Aspirant. Vol. 12.

Represented (Measured) By An Alternation Of The First Prime Numbers. GJPAM. 2015;**11**(4):

2018. pp. S.109-S.116

1803-1818

[2] Vinogradov IM. Osnovy teorii chisel.

Numbers]. Moscow: Gostekhizdat; 1952.

[3] Ozhigova YP. Razvitie teorii chisel v Rossii. [Development of A Theory of Numbers in Russia]. 2nd ed. Moscow: Directory of the Ukrainian SSR; Vol. 38.

[4] Serpinsky V. Chto my znaem i chego ne znaem o prostyh chislah. [What Do We Know and What We Don't Know About Simple Numbers]. Moscow: State Publishing House for physical and mathematical Literature; 1963.

[5] Aleksandrov PS. Jenciklopedija jelementarnoj matematiki/ kniga pervaja

arifmetika// [Encyclopedia of Elementary Mathematics. Volume 1

tehnikoteoreticheskoj literatury;

[6] Prahar K. Raspredelenie prostyh chisel. Izdatel'stvo"MIR"; Moscow.

[7] Shherbakov AG. O raspredelenii prostyh chisel. [On Distribution of Prime Numbers]. Vol. 10. Perspektivy nauki. - Tambov, Izdatel'stvo MOO Fond razvitija nauki i kul'tury; 2013.

[8] Shcherbakov AG. Chetyre prostykh chisla mezhdu kvadratami dvukh posledovatelnykh prostykh chisel. In: NAUKA SEGODNYa globalnyye vyzovy i

Mezhdunarodnoy nauchno-prakticheskoy konferentsii. Vologda: Disput; 2018.

mekhanizmy razvitiya: materialy

pp. S142-S147. [in Russian]

Moscow. 1951. pp. 255-282. [in Russian]

Arithmetic]. Izdatel'stvo

[Fundamentals of A Theory of

*Number Theory and Its Applications*

## Moments of Catalan Triangle Numbers

*Pedro J. Miana and Natalia Romero*

#### **Abstract**

In this chapter, we consider the Catalan numbers, *Cn* <sup>¼</sup> <sup>1</sup> *n*þ1 2*n n* � �, and two of their generalizations, Catalan triangle numbers, *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*, for *n*, *k*∈ . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*kmBj <sup>n</sup>*,*<sup>k</sup>*, P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>1</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup> <sup>n</sup>*,*<sup>k</sup>*, for *j*, *n* ∈ and *m* ∈ ∪ f g0 . We present their closed expressions for some values of *m* and *j*. Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

**Keywords:** Catalan numbers, combinatorial identities, binomial coefficients, moments

#### **1. Introduction**

After the binomial coefficients, the well-known Catalan numbers ð Þ *Cn <sup>n</sup>* <sup>≥</sup><sup>0</sup> are the most frequently occurring combinatorial numbers. They are treated deeply in many books, monographs, and papers (e.g., [1–20]). Catalan numbers play an important role and have a major importance in computer science and combinatorics.

They appear in studying astonishingly many combinatorial problems. They count the number of different ways to triangulate a regular polygon with *n* þ 2 sides; or, the number of ways that 2*n* people seat around a circular table are simultaneously shaking hands with another person at the table in such a way that none of the arms cross each other, and also in tree enumeration problem, see these examples and others in [19, 20].

Other applications of the Catalan numbers appear in engineering in the field of cryptography to form keys for secure transfer of information; in computational geometry, they are generally used in geometric modeling; they may be also found in geographic information systems, geodesy, or medicine.

There are several ways to define Catalan numbers; one of them is recursively by *<sup>C</sup>*<sup>0</sup> <sup>¼</sup> 1 and *Cn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup>*CiCn*�1�*<sup>i</sup>* for *<sup>n</sup>* <sup>≥</sup>1; the first terms in this sequence are

$$\begin{array}{cccc} \text{1, 1, 2, 5, 14, 42, 132, \dots} & \text{(1)} \end{array}$$

In this paper, our main objective is to study in detail the moments of Catalan

*k*¼1

for *j*, *n* ∈ and *m* ∈ ∪ f g0 . In previous papers, the authors have considered some particular cases of these sums: for *j* ¼ 1 and *m* ¼ 0 in [14], for *j* ¼ 2 in [12, 13], and for *j* ¼ 3 and *m* ¼ 0 in [22]. In [7], the authors solved a conjecture posed in [22] about divisibility properties in the case *m* ¼ 0. However, there are no results in the literatures for moments for *j*> 2. We complete and present a full treatment of these moments, for *j* ¼ 1 in Section 2 and for *j* ¼ 2 and for some cases of *j* ¼ 3 in Section 4. In the paper [23], the authors treat several families of binomial sum identities whose definition involves the absolute value function. Here we present alternating sums of for several powers of Catalan triangle numbers (Theorem 2.2, Proposition 4.1 (iii), and Proposition 4.4 (iii)). In ([24], Theorem 2.3), the following identityis

ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup>*

*<sup>n</sup>*,*<sup>k</sup>* ¼ *n n*ð Þ � 2 ð Þ 2*n* � 1 *Cn*�1, *n*≥1*:* (9)

*k*2 *A j*

*<sup>n</sup>*,*<sup>k</sup>* for *j*∈f g 1, 2, 3, 4, 5 ,

*n*∈ , (10)

*<sup>n</sup>*, (11)

*<sup>n</sup>:* (12)

*<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

*n* � 1 þ *k n* � 1 � �<sup>2</sup>

> 2 � �<sup>2</sup>

*C*2

*<sup>n</sup>*,*<sup>k</sup>*, (8)

*<sup>n</sup>*,*<sup>k</sup>*, <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

X*n k*¼1

*kmB <sup>j</sup>*

triangle numbers:

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

proved:

X*n k*¼1

In this paper, we treat P*<sup>n</sup>*

about these natures of the sums.

*n* þ *k n* � �<sup>2</sup>

ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> where

*a n*ð Þ <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

*k*¼0

(Theorem 3.4).

**73**

for third order, we present that

ð Þ �<sup>1</sup> *<sup>k</sup> k*2 *B*2

*<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

and we conjecture some divisibility properties in Conjecture 5.7.

*k*2 *B j*

*<sup>n</sup>*,*<sup>k</sup>* and <sup>P</sup>*<sup>n</sup>*þ<sup>1</sup>

The WZ theory is a powerful tool to show hypergeometric identities. We have applied this tool in Theorem 2.1 to check certain identities. In detail, we have used the Maple program and the EKHAD package as software for the WZ method; see ([25], Example 7.5.3). Although analytic proofs are not presented, alternative proofs as to apply WZ theory [26, 27] or some mathematical software indicate us what these identities hold. Note that an analytic proof will give us some extra information

In Section 3, we prove new identities involving sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and

*k*¼0

2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *a n*ð Þ� *na n*ð Þ¼ � <sup>1</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *b n*ð Þ� <sup>þ</sup> <sup>1</sup> *nb n*ð Þ¼ <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � �*C*<sup>2</sup>

Lemma 3.3 shows that sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>1</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are deeply connected with Catalan numbers. Recurrence relations (30) and (36) (and polynomials in these relations) play delicate roles which allow to give proof of the identity:

In Section 4, we give the moments of second order in Theorem 4.2 and 4.3, and

ð Þ ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> <sup>¼</sup> 4 3ð Þ *a n*ð Þ� � <sup>1</sup> <sup>2</sup>*b n*ð Þ , *<sup>n</sup>* <sup>≥</sup>1, (13)

and Catalan numbers ð Þ *Cn <sup>n</sup>* <sup>≥</sup><sup>0</sup>. In Theorems 3.1 and 3.2, we show that for *n* ≥1,

*n* � *k n*

, *b n*ð Þ <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

The generating formula for Catalan numbers is

$$C(\mathbf{x}) := \frac{\mathbf{1} - \sqrt{\mathbf{1} - 4\mathbf{x}}}{2\mathbf{x}} = \sum\_{n \ge 0} C\_n \mathbf{x}^n, \qquad \mathbf{0} < \mathbf{x} < \mathbf{1}/14 \tag{2}$$

[10] and ([20], Proposition 1.3.1).

Catalan triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*,*k*≥<sup>1</sup> and ð Þ *An*,*<sup>k</sup> <sup>n</sup>*,*k*≥<sup>1</sup> are defined by

$$B\_{n,k} := \frac{k}{n} \binom{2n}{n-k}, \qquad A\_{n,k} := \frac{2k-1}{2n+1} \binom{2n+1}{n+1-k} \quad n,k \in \mathbb{N}, \ k \le n+1. \tag{3}$$

Notice that *Bn*,1 ¼ *An*,1 ¼ *Cn*. In [14], Shapiro introduced Catalan triangles whose entries are given by the coefficients

$$\sum\_{n\geq k} B\_{n,k} \mathfrak{x}^n = \mathfrak{x}^k \mathbf{C}^{2k}(\mathfrak{x}),\tag{4}$$

see a more general approach in [10].

Although the numbers *Bn*,*<sup>k</sup>* (and also *An*,*<sup>k</sup>*) are not as well-known as Catalan numbers, they have also several applications, for example, *Bn*,*<sup>k</sup>* is the number of walks of *n* steps, each in direction *N*, *S*, *W*, or *E*, starting at the origin, remaining in the upper half-plane and ending at height *k*; see more details in [4, 13, 14, 16] for additional information.

Both Catalan triangle numbers may be written in unified expression. We consider combinatorial numbers ð Þ *Cm*,*<sup>k</sup> <sup>m</sup>* <sup>≥</sup>1,*k*≥<sup>0</sup>, given by

$$C\_{m,k} := \frac{m-2k}{m} \binom{m}{k} . \tag{5}$$

These combinatorial numbers ð Þ *Cm*,*<sup>k</sup> <sup>m</sup>* <sup>≥</sup>1,*k*≥<sup>0</sup> are suitable rearrangements of the known ballot numbers ð Þ *am*,*<sup>k</sup>* with *am*,*<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*þ<sup>1</sup> *m*þ1 2*m* � *k m* � � for *<sup>m</sup>* <sup>≥</sup> 0 and 0 <sup>≤</sup>*k*<sup>≤</sup> *<sup>m</sup>*, i.e.,

$$a\_{m,k} = \mathbb{C}\_{2m+1-k, m-k}, \qquad \mathbb{C}\_{m,k} = a\_{m-k-1, m-2k-1} \tag{6}$$

see example [21]. Note that *C*2*n*,*n*�*<sup>k</sup>* ¼ *Bn*,*<sup>k</sup>* and also *C*2*n*þ1,*n*þ1�*<sup>k</sup>* ¼ *An*,*<sup>k</sup>*. In ([9], Theorem 1.1), the authors show that any binomial coefficient can be written as weighted sums along the rows of the Catalan triangle, i.e.,

$$
\binom{n+k+1}{k} = \sum\_{j=0}^{k} \mathcal{C}\_{nj} 2^{k-j}.\tag{7}
$$

The generalized *k*th Catalan numbers *kCn* ≔ <sup>1</sup> *n nk n* � 1 � �, *<sup>k</sup>*≥1, are presented in [17] to count the number of ways of subdividing a convex polygon into *k* disjoint ð Þ *n* þ 1 -polygons by means of nonintersecting diagonals, *k*≥ 1; see also [2, 11].

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

There are several ways to define Catalan numbers; one of them is recursively by

*n*≥0

*<sup>i</sup>*¼<sup>0</sup>*CiCn*�1�*<sup>i</sup>* for *<sup>n</sup>* <sup>≥</sup>1; the first terms in this sequence are

2*n* þ 1 *n* þ 1 � *k* � �

*C*<sup>2</sup>*<sup>k</sup>*

1, 1, 2, 5, 14, 42, 132, … (1)

*Cnxn*, 0< *x*<1*=*14 (2)

*n*, *k*∈ , *k*≤ *n* þ 1*:* (3)

ð Þ *x* , (4)

*:* (5)

for *m* ≥ 0 and 0 ≤*k*≤ *m*,

*:* (7)

, *k*≥1, are presented in

*<sup>C</sup>*<sup>0</sup> <sup>¼</sup> 1 and *Cn* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*�<sup>1</sup>

*Number Theory and Its Applications*

*Bn*,*<sup>k</sup>* <sup>≔</sup> *<sup>k</sup> n*

additional information.

i.e.,

**72**

The generating formula for Catalan numbers is

*C x*ð Þ <sup>≔</sup> <sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi

[10] and ([20], Proposition 1.3.1).

whose entries are given by the coefficients

see a more general approach in [10].

sider combinatorial numbers ð Þ *Cm*,*<sup>k</sup> <sup>m</sup>* <sup>≥</sup>1,*k*≥<sup>0</sup>, given by

weighted sums along the rows of the Catalan triangle, i.e.,

The generalized *k*th Catalan numbers *kCn* ≔ <sup>1</sup>

*n* þ *k* þ 1 *k* � �

known ballot numbers ð Þ *am*,*<sup>k</sup>* with *am*,*<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*þ<sup>1</sup>

2*n n* � *k* � � <sup>1</sup> � <sup>4</sup>*<sup>x</sup>* <sup>p</sup> <sup>2</sup>*<sup>x</sup>* <sup>¼</sup> <sup>X</sup>

, *An*,*<sup>k</sup>* <sup>≔</sup> <sup>2</sup>*<sup>k</sup>* � <sup>1</sup>

X *n* ≥*k*

Catalan triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*,*k*≥<sup>1</sup> and ð Þ *An*,*<sup>k</sup> <sup>n</sup>*,*k*≥<sup>1</sup> are defined by

2*n* þ 1

Notice that *Bn*,1 ¼ *An*,1 ¼ *Cn*. In [14], Shapiro introduced Catalan triangles

*Bn*,*kx<sup>n</sup>* <sup>¼</sup> *<sup>x</sup><sup>k</sup>*

Although the numbers *Bn*,*<sup>k</sup>* (and also *An*,*<sup>k</sup>*) are not as well-known as Catalan numbers, they have also several applications, for example, *Bn*,*<sup>k</sup>* is the number of walks of *n* steps, each in direction *N*, *S*, *W*, or *E*, starting at the origin, remaining in the upper half-plane and ending at height *k*; see more details in [4, 13, 14, 16] for

Both Catalan triangle numbers may be written in unified expression. We con-

*m*

These combinatorial numbers ð Þ *Cm*,*<sup>k</sup> <sup>m</sup>* <sup>≥</sup>1,*k*≥<sup>0</sup> are suitable rearrangements of the

*m*þ1

see example [21]. Note that *C*2*n*,*n*�*<sup>k</sup>* ¼ *Bn*,*<sup>k</sup>* and also *C*2*n*þ1,*n*þ1�*<sup>k</sup>* ¼ *An*,*<sup>k</sup>*. In ([9], Theorem 1.1), the authors show that any binomial coefficient can be written as

> <sup>¼</sup> <sup>X</sup> *k*

[17] to count the number of ways of subdividing a convex polygon into *k* disjoint ð Þ *n* þ 1 -polygons by means of nonintersecting diagonals, *k*≥ 1; see also [2, 11].

*j*¼0

*m k* � �

2*m* � *k m* � �

*am*,*<sup>k</sup>* ¼ *C*2*m*þ1�*k*,*m*�*<sup>k</sup>*, *Cm*,*<sup>k</sup>* ¼ *am*�*k*�1,*m*�2*k*�1, (6)

*Cn*,*j*2*<sup>k</sup>*�*<sup>j</sup>*

*nk n* � 1 � �

*n*

*Cm*,*<sup>k</sup>* <sup>≔</sup> *<sup>m</sup>* � <sup>2</sup>*<sup>k</sup>*

In this paper, our main objective is to study in detail the moments of Catalan triangle numbers:

$$\sum\_{k=1}^{n} k^{m} B\_{n,k}^{j}, \qquad \sum\_{k=1}^{n+1} (2k-1)^{m} A\_{n,k}^{j}, \tag{8}$$

for *j*, *n* ∈ and *m* ∈ ∪ f g0 . In previous papers, the authors have considered some particular cases of these sums: for *j* ¼ 1 and *m* ¼ 0 in [14], for *j* ¼ 2 in [12, 13], and for *j* ¼ 3 and *m* ¼ 0 in [22]. In [7], the authors solved a conjecture posed in [22] about divisibility properties in the case *m* ¼ 0. However, there are no results in the literatures for moments for *j*> 2. We complete and present a full treatment of these moments, for *j* ¼ 1 in Section 2 and for *j* ¼ 2 and for some cases of *j* ¼ 3 in Section 4.

In the paper [23], the authors treat several families of binomial sum identities whose definition involves the absolute value function. Here we present alternating sums of for several powers of Catalan triangle numbers (Theorem 2.2, Proposition 4.1 (iii), and Proposition 4.4 (iii)). In ([24], Theorem 2.3), the following identityis proved:

$$\sum\_{k=1}^{n}(-1)^{k}k^{2}B\_{n,k}^{2} = n(n-2)(2n-1)C\_{n-1}, \qquad n \ge 1. \tag{9}$$

In this paper, we treat P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup> k*2 *B j <sup>n</sup>*,*<sup>k</sup>* and <sup>P</sup>*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup> k*2 *A j <sup>n</sup>*,*<sup>k</sup>* for *j*∈f g 1, 2, 3, 4, 5 , and we conjecture some divisibility properties in Conjecture 5.7.

The WZ theory is a powerful tool to show hypergeometric identities. We have applied this tool in Theorem 2.1 to check certain identities. In detail, we have used the Maple program and the EKHAD package as software for the WZ method; see ([25], Example 7.5.3). Although analytic proofs are not presented, alternative proofs as to apply WZ theory [26, 27] or some mathematical software indicate us what these identities hold. Note that an analytic proof will give us some extra information about these natures of the sums.

In Section 3, we prove new identities involving sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> where

$$a(n) := \sum\_{k=0}^{n} \binom{n+k}{n}^2, \qquad b(n) := \sum\_{k=0}^{n} \frac{n-k}{n} \binom{n-1+k}{n-1}^2 \qquad n \in \mathbb{N}, \tag{10}$$

and Catalan numbers ð Þ *Cn <sup>n</sup>* <sup>≥</sup><sup>0</sup>. In Theorems 3.1 and 3.2, we show that for *n* ≥1,

$$(2(2n+1)a(n) - na(n-1)) \ = (21n+8) \left(\frac{n+1}{2}\right)^2 C\_n^2,\tag{11}$$

$$2(2n+1)b(n+1) - nb(n) = \left(7n^2 + 8n + 2\right)C\_n^2. \tag{12}$$

Lemma 3.3 shows that sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>1</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are deeply connected with Catalan numbers. Recurrence relations (30) and (36) (and polynomials in these relations) play delicate roles which allow to give proof of the identity:

$$\left( (n+1)\mathbb{C}\_n \right)^2 = 4(\mathfrak{A}(n-1) - \mathfrak{B}(n)), \qquad n \ge 1,\tag{13}$$

(Theorem 3.4).

In Section 4, we give the moments of second order in Theorem 4.2 and 4.3, and for third order, we present that

$$\sum\_{k=0}^{n} B\_{n,k}^{3} = \frac{n+1}{2} \mathbb{C}\_{n} b(n), \qquad \sum\_{k=1}^{n+1} A\_{n,k}^{3} = (n+1) \mathbb{C}\_{n} \left( \left( 2(n+1) \mathbb{C}\_{n} \right)^{2} - 3a(n) \right), \tag{14}$$

for *n*≥ 1; see also ([22], Section 3).

Finally, we conjecture some divisibility properties in Section 5; in particular

$$\sum\_{k=1}^{n} k^{2m} B\_{n,k} = \frac{n+1}{2} C\_n n P\_{m-1}(n),\tag{15}$$

which appears in a problem related with the dynamical behavior of a family of iterative processes has been proved in ([8], Theorem 5). These numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥*k*≥<sup>1</sup> have been analyzed in many ways. For instance, symmetric functions have been used in [1], recurrence relations in [15], or in [6] the Newton interpolation formula, which is applied to conclude divisibility properties of the sums of products of

> 2*n* þ 1 *n* þ 1 � *k*

which is considered in [13]. Notice that *An*,1 ¼ *Cn* and *C*2*n*þ1,*n*�*k*þ<sup>1</sup> ¼ *An*,*<sup>k</sup>* for

Entries *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>* of the above two particular Catalan triangles satisfy the

For *m* ∈ ∪ f g0 , we define the moments of order *m* by the sum

theorem. We apply the WZ theory to show the following moments for

<sup>2</sup> *Cn:*

**Theorem 2.1.** *For n*∈ , *the following identities hold*:

*n* þ 1 <sup>2</sup> *Cn*,

*<sup>k</sup>mBn*,*<sup>k</sup>*, <sup>Λ</sup>*m*ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

*Bn*,*<sup>k</sup>* ¼ *Bn*�1,*k*�<sup>1</sup> þ 2*Bn*�1,*<sup>k</sup>* þ *Bn*�1,*k*þ1, *k*≥2, (24)

*An*,*<sup>k</sup>* ¼ *An*�1,*k*�<sup>1</sup> þ 2*An*�1,*<sup>k</sup>* þ *An*�1,*k*þ1, *k*≥2*:* (25)

ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mAn*,*<sup>k</sup>*, *<sup>n</sup>*<sup>≥</sup> <sup>1</sup>*:* (26)

*k*¼0

As it was shown in [14], the values of the sums (or moments of order 0) of *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>* are expressed in terms of Catalan numbers; see item (i) and (iii) in the next

� �, *<sup>n</sup>*, *<sup>k</sup>*<sup>∈</sup> , *<sup>k</sup>*<sup>≤</sup> *<sup>n</sup>* <sup>þ</sup> 1, (22)

ð23Þ

Other combinatorial numbers *An*,*<sup>k</sup>* defined as follows

2*n* þ 1

appear as the entries of this other Catalan triangle,

*An*,*<sup>k</sup>* <sup>≔</sup> <sup>2</sup>*<sup>k</sup>* � <sup>1</sup>

binomial coefficients.

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

*k*≤*n* þ 1.

and

recurrence relations

<sup>Δ</sup>*m*ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

*m* ∈ f g 0, 1, … 7 .

i*:* Δ0ð Þ¼ *n*

**75**

Δ2ð Þ¼ *n n*

*k*¼0

*n* þ 1 <sup>2</sup> *Cn*,

Δ4ð Þ¼ *n n*ð Þ 2*n* � 1

*n* þ 1 <sup>2</sup> *Cn*,

<sup>Δ</sup>6ð Þ¼ *<sup>n</sup> <sup>n</sup>* <sup>6</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>4</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � � *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

$$\sum\_{k=1}^{n} k^{2m-1} B\_{n,k} = 2^{n-m-1} Q\_{m-1}(n),\tag{16}$$

$$\sum\_{k=1}^{n+1} k^{2m} A\_{n,k} = (n+1) \mathbf{C}\_n \mathbf{R}\_{m-1}(n),\tag{17}$$

$$\sum\_{k=1}^{n+1} k^{2m-1} A\_{n,k} = 2^{2n} \mathbb{S}\_{m-1}(n),\tag{18}$$

where *Pm*�1, *Qm*�1, *Rm*�<sup>1</sup> and *Sm*�<sup>1</sup> are polynomials of integer coefficients at the degree at most *m* � 1 (Conjectures 5.1 and 5.2). In Conjecture 5.3, we state that the factor *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn* could divide <sup>P</sup>*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*mB*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ ; similarly the factor ð Þ *n* þ 1 *Cn* might divide P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*mA*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n* ∈ (Conjecture 5.4). Similar conjectures about moments of fourth order and alternating sums are also presented in Conjectures 5.5–5.7.

#### **2. Sums and alternating sums of Catalan triangle numbers**

Catalan triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥1,1≤*k*≤*<sup>n</sup>* were introduced in [14]. These combinatorial numbers *Bn*,*<sup>k</sup>* are the entries of the following Catalan triangle:


which are given by

$$B\_{n,k} := \frac{k}{n} \binom{2n}{n-k}, \ n, k \in \mathbb{N}, \ k \le n. \tag{20}$$

Notice that *Bn*,1 ¼ *Cn* and *Bn*,*<sup>n</sup>* ¼ 1 *n*≥ 1.

In the last years, Catalan triangle (19) has been studied in detail. For instance, the formula

$$\sum\_{k=1}^{i} B\_{n,k} B\_{n,n+k-i}(n+2k-i) = (n+1)C\_n \binom{2(n-1)}{i-1}, \quad i \le n,\tag{21}$$

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

which appears in a problem related with the dynamical behavior of a family of iterative processes has been proved in ([8], Theorem 5). These numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥*k*≥<sup>1</sup> have been analyzed in many ways. For instance, symmetric functions have been used in [1], recurrence relations in [15], or in [6] the Newton interpolation formula, which is applied to conclude divisibility properties of the sums of products of binomial coefficients.

Other combinatorial numbers *An*,*<sup>k</sup>* defined as follows

$$A\_{n,k} := \frac{2k-1}{2n+1} \binom{2n+1}{n+1-k}, \quad n,k \in \mathbb{N}, \ k \le n+1,\tag{22}$$

appear as the entries of this other Catalan triangle,


which is considered in [13]. Notice that *An*,1 ¼ *Cn* and *C*2*n*þ1,*n*�*k*þ<sup>1</sup> ¼ *An*,*<sup>k</sup>* for *k*≤*n* þ 1.

Entries *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>* of the above two particular Catalan triangles satisfy the recurrence relations

$$B\_{n,k} = B\_{n-1,k-1} + 2B\_{n-1,k} + B\_{n-1,k+1}, \qquad k \ge 2,\tag{24}$$

and

X*n k*¼0 *B*3

factor *<sup>n</sup>*þ<sup>1</sup>

might divide P*<sup>n</sup>*þ<sup>1</sup>

Conjectures 5.5–5.7.

which are given by

the formula

**74**

X *i*

*k*¼1

*<sup>n</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2

*Number Theory and Its Applications*

*Cnb n*ð Þ, <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

X*n k*¼1

X*n k*¼1

X*n*þ1 *k*¼1

> X*n*þ1 *k*¼1

*k*<sup>2</sup>*m*�<sup>1</sup>

*<sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*mB*<sup>3</sup>

**2. Sums and alternating sums of Catalan triangle numbers**

natorial numbers *Bn*,*<sup>k</sup>* are the entries of the following Catalan triangle:

*Bn*,*<sup>k</sup>* <sup>≔</sup> *<sup>k</sup> n*

*Bn*,*kBn*,*n*þ*k*�*<sup>i</sup>*ð Þ¼ *n* þ 2*k* � *i* ð Þ *n* þ 1 *Cn*

Notice that *Bn*,1 ¼ *Cn* and *Bn*,*<sup>n</sup>* ¼ 1 *n*≥ 1.

2*n n* � *k* � �

In the last years, Catalan triangle (19) has been studied in detail. For instance,

about moments of fourth order and alternating sums are also presented in

for *n*≥ 1; see also ([22], Section 3).

<sup>2</sup> *Cn* could divide <sup>P</sup>*<sup>n</sup>*

*<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*mA*<sup>3</sup>

*k*¼1 *A*3

*k*<sup>2</sup>*m*�<sup>1</sup>

Finally, we conjecture some divisibility properties in Section 5; in particular

*Bn*,*<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*<sup>n</sup>*�*m*�<sup>1</sup>

where *Pm*�1, *Qm*�1, *Rm*�<sup>1</sup> and *Sm*�<sup>1</sup> are polynomials of integer coefficients at the degree at most *m* � 1 (Conjectures 5.1 and 5.2). In Conjecture 5.3, we state that the

Catalan triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥1,1≤*k*≤*<sup>n</sup>* were introduced in [14]. These combi-

*<sup>k</sup>*<sup>2</sup>*mBn*,*<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>n</sup>*,*<sup>k</sup>* <sup>¼</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* ð Þ <sup>2</sup>ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> � <sup>3</sup>*a n*ð Þ

*<sup>k</sup>*<sup>2</sup>*mAn*,*<sup>k</sup>* <sup>¼</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *CnRm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (17)

*An*,*<sup>k</sup>* <sup>¼</sup> <sup>2</sup><sup>2</sup>*nSm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (18)

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ ; similarly the factor ð Þ *n* þ 1 *Cn*

, *n*, *k*∈ , *k*≤*n:* (20)

, *i*≤*n*, (21)

2ð Þ *n* � 1 *i* � 1 � �

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n* ∈ (Conjecture 5.4). Similar conjectures

� �

<sup>2</sup> *CnnPm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (15)

*Qm*�<sup>1</sup>ð Þ *n* , (16)

, (14)

ð19Þ

$$A\_{n,k} = A\_{n-1,k-1} + 2A\_{n-1,k} + A\_{n-1,k+1}, \qquad k \ge 2. \tag{25}$$

For *m* ∈ ∪ f g0 , we define the moments of order *m* by the sum

$$\Delta\_m(n) := \sum\_{k=0}^n k^m B\_{n,k}, \qquad \qquad \Lambda\_m(n) := \sum\_{k=0}^{n+1} (2k-1)^m A\_{n,k}, \qquad n \ge 1. \tag{26}$$

As it was shown in [14], the values of the sums (or moments of order 0) of *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>* are expressed in terms of Catalan numbers; see item (i) and (iii) in the next theorem. We apply the WZ theory to show the following moments for *m* ∈ f g 0, 1, … 7 .

**Theorem 2.1.** *For n*∈ , *the following identities hold*:

$$\begin{aligned} \text{i. } \Delta\_0(n) &= \frac{n+1}{2} C\_n, \\ \Delta\_2(n) &= n \frac{n+1}{2} C\_n, \\ \Delta\_4(n) &= n(2n-1) \frac{n+1}{2} C\_n, \\ \Delta\_6(n) &= n \left( 6n^2 + 4n + 1 \right) \frac{n+1}{2} C\_n. \end{aligned}$$

$$\begin{aligned} \text{i.i. } \Lambda\_1(n) &= 2^{2n-2}, \\ \Lambda\_3(n) &= 2^{2n-3}(3n-1), \\ \Lambda\_5(n) &= 2^{2n-4}(15n(n-1)+2), \\ \Lambda\_7(n) &= 2^{2n-5}(105n^3 - 210n^2 + 147n - 34). \end{aligned}$$

$$\begin{aligned} \text{iii. } \Lambda\_0(n) &= (n+1)\mathbf{C}\_n, \\ \Lambda\_2(n) &= (n+1)\mathbf{C}\_n(4n+1), \\ \Lambda\_4(n) &= (n+1)\mathbf{C}\_n(32n^2 + 8n+1), \\ \Lambda\_6(n) &= (n+1)\mathbf{C}\_n(384n^3 - 32n^2 + 12n+1). \end{aligned}$$

$$\begin{aligned} \text{iv. } \Lambda\_1(n) &= 2^{2n}, \\ \Lambda\_3(n) &= 2^{2n}(6n+1), \\ \Lambda\_5(n) &= 2^{2n}(60n^2 + 1), \end{aligned}$$

$$
\Lambda\_7(n) = 2^{2n} \left( 840n^3 - 420n^2 + 126n + 1 \right).
$$

For alternating sums, the following theorem was proved in [5] and ([22], Corollary 1.3).

**Theorem 2.2.** *For n*≥ 1, *we have*

$$\begin{aligned} \text{i. } &\sum\_{k=1}^{n} \left(-\mathbf{1}\right)^{k} B\_{n,k} = -C\_{n-1}, \\\\ \text{ii. } &\sum\_{k=1}^{n+1} \left(-\mathbf{1}\right)^{k} A\_{n,k} = \mathbf{0}. \end{aligned}$$

Other interesting combinatorial numbers which have been deeply studied in the last decade are the well-known harmonic numbers ð Þ *Hn <sup>n</sup>*≥<sup>1</sup>. These numbers are given by the following formula:

$$H\_n = \sum\_{k=1}^n \frac{1}{k}, \quad n \in \mathbb{N}.\tag{27}$$

X*n*�1 *k*¼0 *B*2

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

lowing recurrence relation:

where polynomials *pi*

of Catalan numbers and ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup>.

ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>29</sup> *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

<sup>¼</sup> 8 21 ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>29</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup>

<sup>¼</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

<sup>¼</sup> 2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>3</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

2 � �<sup>2</sup>

*C*2

2 � �<sup>2</sup>

that 29 <sup>¼</sup> ð Þ <sup>21</sup> � <sup>1</sup> <sup>þ</sup> <sup>8</sup> *<sup>C</sup>*<sup>2</sup>

recurrence (30) to get that

that

**77**

**3. Sums of squares of combinatorial numbers**

*a n*ð Þ <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

� �

**Theorem 3.1.** *For n* ≥1*, the following identity holds*

We consider the sequence of integer numbers defined by

*k*¼0

*<sup>n</sup>*,*kHn*�*<sup>k</sup>*, and <sup>X</sup>*<sup>n</sup>*

*n* þ *k n* � �<sup>2</sup>

*<sup>i</sup>*∈f g 1,2,3 are defined by

*<sup>p</sup>*3ð Þ *<sup>n</sup>* <sup>≔</sup> � <sup>4</sup>ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> <sup>2</sup>

2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *a n*ð Þ� *na n*ð Þ¼ � <sup>1</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>p</sup>*1ð Þ *<sup>n</sup>* <sup>≔</sup> 2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ð Þ <sup>21</sup>*<sup>n</sup>* � <sup>13</sup> *<sup>n</sup>*<sup>2</sup>

Next, in the following theorem, we provide an identity which relates the square

*Proof.* We show this identity by induction method. For *n* ¼ 1, we check directly

<sup>¼</sup> 4 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

*a n*ð Þþ *p*3ð Þ *n* þ 1 *a n*ð Þ � 1

ð Þ 2 2ð Þ *n* þ 3 *a n*ð Þ� þ 1 ð Þ *n* þ 1 *a n*ð Þ ,

� �*a n*ð Þ

*a n*ð Þ� <sup>þ</sup> <sup>1</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup>

where we have applied the induction hypothesis. Then we apply the law of

*Cn*þ<sup>1</sup>

<sup>¼</sup> *<sup>p</sup>*1ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *a n*ð Þþ <sup>þ</sup> <sup>1</sup> 8 21 ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>29</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup> � *<sup>p</sup>*2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

Note that *a*ð Þ¼ 0 1, *a*ð Þ¼ 1 5, *a*ð Þ¼ 2 46, *a*ð Þ¼ 3 517, *a*ð Þ¼ 4 6376, etc. This sequence appears indexed in the On-Line Encyclopedia of Integer Sequences by N.J. A. Sloane [16] with the reference *A*112029. V. Kotesovec in 2012 proved the fol-

*k*¼1 *A*2

*p*1ð Þ *n a n*ð Þ¼ *p*2ð Þ *n a n*ð Þþ � 1 *p*3ð Þ *n a n*ð Þ � 2 , *n*≥2, (30)

*<sup>p</sup>*2ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>1365</sup>*n*<sup>4</sup> � <sup>1517</sup>*n*<sup>3</sup> <sup>þ</sup> <sup>240</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>216</sup>*<sup>n</sup>* � 64, (32)

1. Now suppose that the identity holds for any *m* ≤*n*. Note

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>21</sup>*<sup>n</sup>* <sup>þ</sup> <sup>8</sup> 4 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>n</sup>*,*kHn*�*k*þ<sup>1</sup>*:* (28)

, *n*∈ ∪ f g0 *:* (29)

, (31)

ð Þ 21*n* þ 8 *:* (33)

*C*2

2 � �<sup>2</sup>

ð Þ 2 2ð Þ *n* þ 1 *a n*ð Þ� *na n*ð Þ � 1 ,

*C*2 *n*

*a n*ð Þ

*<sup>n</sup>:* (34)

2 � �<sup>2</sup>

A deep treatment of closed formulas for the sums of the form P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*akHk* is given in [18]. Also, the WZ theory is applied to get identities in [26], and infinite series involving harmonic numbers is presented in [3]. See other approaches in ([28], Chapter 7) and reference therein.

In ([22], Corollary 1.5) the next relationships between Catalan triangle numbers and harmonic numbers ð Þ *Hn <sup>n</sup>*≥<sup>1</sup> are given.

**Corollary 2.3.** *For n* ≥1, *we have*

$$\begin{aligned} \text{i. } \sum\_{k=0}^{n-1} B\_{n,k} H\_{n-k} &= \frac{(2nH\_n - 1)(n+1)}{4n} \mathbf{C}\_n - \frac{2^{2n-1} - 1}{2n}, \\\\ \text{ii. } \sum\_{k=1}^n A\_{n,k} H\_{n-k+1} &= H\_n (n+1) \mathbf{C}\_n - \frac{2^{2n} - 1}{2n+1}. \end{aligned}$$

**Remark.** It is worth to consider other powers of Catalan triangle numbers and harmonic numbers to obtain, for example, formulae of

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

ii*:* <sup>Δ</sup>1ð Þ¼ *<sup>n</sup>* 22*<sup>n</sup>*�<sup>2</sup>

<sup>Δ</sup>3ð Þ¼ *<sup>n</sup>* 22*<sup>n</sup>*�<sup>3</sup>

*Number Theory and Its Applications*

iii*:* Λ0ð Þ¼ *n* ð Þ *n* þ 1 *Cn*,

iv*:* <sup>Λ</sup>1ð Þ¼ *<sup>n</sup>* <sup>2</sup><sup>2</sup>*<sup>n</sup>*,

Corollary 1.3).

i. <sup>P</sup>*<sup>n</sup> k*¼1

ii. *n* P þ1 *k*¼1

> i. *n* P�1 *k*¼0

ii. <sup>P</sup>*<sup>n</sup> k*¼1

**76**

ð Þ �<sup>1</sup> *<sup>k</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup>*

given by the following formula:

Chapter 7) and reference therein.

and harmonic numbers ð Þ *Hn <sup>n</sup>*≥<sup>1</sup> are given. **Corollary 2.3.** *For n* ≥1, *we have*

*Bn*,*kHn*�*<sup>k</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*nHn*�<sup>1</sup> ð Þ *<sup>n</sup>*þ<sup>1</sup>

*An*,*kHn*�*k*þ<sup>1</sup> <sup>¼</sup> *Hn*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* � <sup>2</sup>2*n*�<sup>1</sup>

harmonic numbers to obtain, for example, formulae of

<sup>Λ</sup>3ð Þ¼ *<sup>n</sup>* <sup>2</sup><sup>2</sup>*<sup>n</sup>*ð Þ <sup>6</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> , <sup>Λ</sup>5ð Þ¼ *<sup>n</sup>* 22*<sup>n</sup>* <sup>60</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �,

**Theorem 2.2.** *For n*≥ 1, *we have*

*Bn*,*<sup>k</sup>* ¼ �*Cn*�1,

*An*,*<sup>k</sup>* ¼ 0.

,

Λ2ð Þ¼ *n* ð Þ *n* þ 1 *Cn*ð Þ 4*n* þ 1 ,

ð Þ 3*n* � 1 , <sup>Δ</sup>5ð Þ¼ *<sup>n</sup>* 22*<sup>n</sup>*�<sup>4</sup>ð Þ <sup>15</sup>*n n*ð Þþ � <sup>1</sup> <sup>2</sup> ,

<sup>Λ</sup>4ð Þ¼ *<sup>n</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>32</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �,

<sup>Δ</sup>7ð Þ¼ *<sup>n</sup>* 22*<sup>n</sup>*�<sup>5</sup> <sup>105</sup>*n*<sup>3</sup> � <sup>210</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>147</sup>*<sup>n</sup>* � <sup>34</sup> � �*:*

<sup>Λ</sup>6ð Þ¼ *<sup>n</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>384</sup>*n*<sup>3</sup> � <sup>32</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>12</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �*:*

<sup>Λ</sup>7ð Þ¼ *<sup>n</sup>* <sup>2</sup><sup>2</sup>*<sup>n</sup>* <sup>840</sup>*n*<sup>3</sup> � <sup>420</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>126</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �*:*

For alternating sums, the following theorem was proved in [5] and ([22],

Other interesting combinatorial numbers which have been deeply studied in the

last decade are the well-known harmonic numbers ð Þ *Hn <sup>n</sup>*≥<sup>1</sup>. These numbers are

*k*¼1

in [18]. Also, the WZ theory is applied to get identities in [26], and infinite series involving harmonic numbers is presented in [3]. See other approaches in ([28],

In ([22], Corollary 1.5) the next relationships between Catalan triangle numbers

<sup>2</sup>*<sup>n</sup>* ,

<sup>2</sup>*n*þ<sup>1</sup> *:*

**Remark.** It is worth to consider other powers of Catalan triangle numbers and

1 *k*

, *n* ∈ *:* (27)

*<sup>k</sup>*¼<sup>1</sup>*akHk* is given

*Hn* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

A deep treatment of closed formulas for the sums of the form P*<sup>n</sup>*

<sup>4</sup>*<sup>n</sup> Cn* � <sup>2</sup>2*n*�1�<sup>1</sup>

$$\sum\_{k=0}^{n-1} B\_{n,k}^2 H\_{n-k}, \qquad \text{and} \qquad \sum\_{k=1}^n A\_{n,k}^2 H\_{n-k+1}. \tag{28}$$

#### **3. Sums of squares of combinatorial numbers**

We consider the sequence of integer numbers defined by

$$a(n) \coloneqq \sum\_{k=0}^{n} \binom{n+k}{n}^2, \qquad n \in \mathbb{N} \cup \{0\}. \tag{29}$$

Note that *a*ð Þ¼ 0 1, *a*ð Þ¼ 1 5, *a*ð Þ¼ 2 46, *a*ð Þ¼ 3 517, *a*ð Þ¼ 4 6376, etc. This sequence appears indexed in the On-Line Encyclopedia of Integer Sequences by N.J. A. Sloane [16] with the reference *A*112029. V. Kotesovec in 2012 proved the following recurrence relation:

$$p\_1(n)a(n) = p\_2(n)a(n-1) + p\_3(n)a(n-2), \qquad n \ge 2,\tag{30}$$

where polynomials *pi* � � *<sup>i</sup>*∈f g 1,2,3 are defined by

$$p\_1(n) := \Im(2n+1)(2\mathbf{1}n - \mathbf{1}\mathfrak{Z})n^2,\tag{31}$$

$$p\_2(n) \coloneqq 1365n^4 - 1517n^3 + 240n^2 + 216n - 64,\tag{32}$$

$$p\_3(n) \coloneqq -4(n-1)(2n-1)^2(2\mathbf{1}n + 8). \tag{33}$$

Next, in the following theorem, we provide an identity which relates the square of Catalan numbers and ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup>.

**Theorem 3.1.** *For n* ≥1*, the following identity holds*

$$(2(2n+1)a(n) - na(n-1) = (21n+8)\left(\frac{n+1}{2}\right)^2 C\_n^2. \tag{34}$$

*Proof.* We show this identity by induction method. For *n* ¼ 1, we check directly that 29 <sup>¼</sup> ð Þ <sup>21</sup> � <sup>1</sup> <sup>þ</sup> <sup>8</sup> *<sup>C</sup>*<sup>2</sup> 1. Now suppose that the identity holds for any *m* ≤*n*. Note that

$$\begin{aligned} (21n+8) \binom{n+2}{2}^2 \mathcal{C}\_{n+1}^2 &= (21n+8)4(2n+1)^2 \binom{n+1}{2}^2 \mathcal{C}\_n^2 \\ &= 4(2n+1)^2 (2(2n+1)a(n) - na(n-1)), \end{aligned}$$

where we have applied the induction hypothesis. Then we apply the law of recurrence (30) to get that

$$\begin{aligned} &(21n+8)(21n+29)\left(\frac{n+2}{2}\right)^2\mathbf{C}\_{n+1} \\ &=8(21n+29)(2n+1)^3a(n)+p\_3(n+1)a(n-1) \\ &=p\_1(n+1)a(n+1)+\left(8(21n+29)(2n+1)^3-p\_2(n+1)\right)a(n) \\ &=2(2n+3)(21n+8)(n+1)^2a(n+1)-(21n+8)(n+1)^3a(n) \\ &=(21n+8)(n+1)^2(2(2n+3)a(n+1)-(n+1)a(n)),\end{aligned}$$

and we conclude the proof. □ Now we consider this second sequence of integer numbers defined by

$$b(n) \coloneqq \sum\_{k=0}^{n} \frac{k}{n} \binom{2n-k-1}{n-1}^2 = \sum\_{k=0}^{n} \frac{n-k}{n} \binom{n-1+k}{n-1}^2, \qquad n \in \mathbb{N}.\tag{35}$$

Note that *b*ð Þ¼ 1 1, *b*ð Þ¼ 2 3, *b*ð Þ¼ 3 19, *b*ð Þ¼ 4 163, *b*ð Þ¼ 5 1625, etc. This sequence also appears indexed in the On-Line Encyclopedia of Integer Sequences by N.J.A. Sloane [16] with the reference *A*183069, and V. Kotesovec proved the following recurrence relation:

$$q\_1(n)b(n) = q\_2(n)b(n-1) + q\_3(n)b(n-2), \qquad n \ge 3,\tag{36}$$

where polynomials *qi* � � *<sup>i</sup>* <sup>∈</sup>f g 1,2,3 are defined by

$$q\_1(n) \coloneqq 2n^2(2n-1)\left(7n^2 - 20n + 14\right),\tag{37}$$

*where Q n*ð Þ <sup>≔</sup> <sup>147</sup>*n*<sup>4</sup> � <sup>546</sup>*n*<sup>3</sup> <sup>þ</sup> <sup>666</sup>*n*<sup>2</sup> � <sup>293</sup>*<sup>n</sup>* <sup>þ</sup> 34.

**Theorem 3.4**. *For n*≥ 1*, the following identity holds*

*Proof.* We write by *c n*ð Þ¼ ð Þ ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> <sup>¼</sup> <sup>2</sup>*<sup>n</sup>*

*p*1ð Þ *n* � 1 *q*1ð Þ *n* 4 3ð Þ *a n*ð Þ� � 1 2*b n*ð Þ

¼ *p*1ð Þ *n* � 1 *q*2ð Þ *n* ð Þ 12*a n*ð Þ� � 2 8*b n*ð Þ � 1 þ*p*1ð Þ *n* � 1 *q*3ð Þ *n* ð Þ 12*a n*ð Þ� � 3 8*b n*ð Þ � 2

By the induction method and Theorem 3.1, we have that

*p*1ð Þ *n* � 1 *q*1ð Þ *n* 4 3ð Þ *a n*ð Þ� � 1 2*b n*ð Þ

<sup>þ</sup>24 2ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup>

<sup>þ</sup>96 2ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup>

for *<sup>n</sup>*<sup>≥</sup> 2. Since 4 2ð Þ *<sup>n</sup>* � <sup>3</sup> <sup>2</sup>

24 2ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup>

**79**

Finally, we get that

<sup>¼</sup> <sup>12</sup>*q*1ð Þ *<sup>n</sup> <sup>p</sup>*2ð Þ *<sup>n</sup>* � <sup>1</sup> *a n*ð Þþ � <sup>2</sup> *<sup>p</sup>*3ð Þ *<sup>n</sup>* � <sup>1</sup> *a n*ð Þ � <sup>3</sup> � � �8*p*1ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>q</sup>*2ð Þ *<sup>n</sup> b n*ð Þþ � <sup>1</sup> *<sup>q</sup>*3ð Þ *<sup>n</sup> b n*ð Þ � <sup>2</sup> � �

<sup>¼</sup> <sup>12</sup>*a n*ð Þð � <sup>2</sup> *<sup>p</sup>*1ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>q</sup>*2ð Þþ *<sup>n</sup>* <sup>16</sup>*Q n*ð Þ <sup>2</sup>ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>3</sup> � �

�8*p*1ð Þð *n* � 1 *q*2ð Þ *n b n*ð Þ� � 1 2*p*1ð Þ *n* � 1 *q*3ð Þ *n b n*ð Þ � 2

<sup>þ</sup>12*a n*ð Þ � <sup>3</sup> *<sup>p</sup>*1ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>q</sup>*3ð Þ� *<sup>n</sup>* <sup>8</sup>*Q n*ð Þð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> ð Þ *<sup>n</sup>* � <sup>2</sup> � �

where we have applied the recurrence relations (30) and (36) and Lemma 3.3.

*p*1ð Þ *n* � 1 *q*1ð Þ *n* 4 3ð *a n*ð Þ� � 1 2*b n*ð ÞÞ ¼ *p*1ð Þ *n* � 1 *q*2ð Þ *n c n*ð Þþ � 1 *p*1ð Þ *n* � 1 *q*3ð Þ *n c n*ð Þ � 2

*c n*ð Þ¼ � <sup>2</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>2</sup>

<sup>¼</sup> *<sup>p</sup>*1ð Þ *<sup>n</sup>* � <sup>1</sup> *c n*ð Þ � <sup>1</sup> *<sup>q</sup>*2ð Þþ *<sup>n</sup> <sup>q</sup>*3ð Þ *<sup>n</sup>* ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>2</sup>

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> k*¼0

2*n n* � �<sup>2</sup>

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

equivalent way.

following identity

Note that

Our last aim of this section is to show an alternative of the following identity

in Theorem 3.4. An original proof is presented in ([22], Theorem 2.3 (ii)), and it is a straightforward consequence of a more general identity in combinatorial numbers ([22], Theorem 2.3 (i)). The proof which we present here allows to recognize the natural connection among the sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> and the Catalan numbers ð Þ *Cn <sup>n</sup>*≥<sup>0</sup>. Note that one may rewrite the identity (43) in an

2*n* � 1 � *k n* � 1 � �<sup>2</sup>

ð Þ ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> <sup>¼</sup> 4 3ð Þ *a n*ð Þ� � <sup>1</sup> <sup>2</sup>*b n*ð Þ , *<sup>n</sup>*≥1*:* (44)

*n* � �<sup>2</sup>

where sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are considered in the second section.

*c n*ð Þ¼ 4 3ð Þ *a n*ð Þ� � 1 2*b n*ð Þ , *n* ≥1, (45)

*Q n*ð Þð Þ 2 2ð Þ *n* � 3 *a n*ð Þ� � 2 ð Þ *n* � 2 *a n*ð Þ � 3 ,

*Q n*ð Þð Þ 21*n* � 34 *c n*ð Þ � 2

4 2ð Þ *<sup>n</sup>* � <sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>3</sup>*Q n*ð Þ !

*Q n*ð Þð Þ 21*n* � 34 *c n*ð Þ¼ � 2 3*p*1ð Þ *n* � 1 *Q n*ð Þ*c n*ð Þ � 1 , *n*≥1*:* (46)

*c n*ð Þ � 1 for *n* ≥2, we have that

, (43)

, and then we have to check the

3*n* � 2*k n*

$$q\_2(n) \coloneqq 455n^5 - 2427n^4 + 4850n^3 - 4406n^2 + 1728n - 216,\tag{38}$$

$$q\_3(n) \coloneqq -4(n-2)(2n-3)^2(7n^2 - 6n + 1). \tag{39}$$

In a similar way, we obtain an identity which relates numbers ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> to the square of Catalan numbers.

**Theorem 3.2.** *For n* ≥1*, the following identity holds*

$$\mathcal{Q}(2n+1)b(n+1) - nb(n) = \left(7n^2 + 8n + 2\right)\mathcal{C}\_n^2. \tag{40}$$

*Proof.* We prove the identity by the induction method. For *n* ¼ 1, we directly check the identity. Suppose that the identity holds for a given number *n*. Since ð Þ *n* þ 2 *Cn*þ<sup>1</sup> ¼ 2 2ð Þ *n* þ 1 *Cn*, we have that

$$\begin{aligned} &\left(7n^2+8n+2\right)\left(7n^2+22n+17\right)(n+2)^2\mathcal{C}\_{n+1}^2\\ &=\left(7n^2+8n+2\right)\left(7n^2+22n+17\right)4\left(2n+1\right)^2\mathcal{C}\_n^2\\ &=4\left(2n+1\right)^2\left(7n^2+22n+17\right)\left(2\left(2n+1\right)b\left(n+1\right)-nb\left(n\right)\right)\\ &=8\left(2n+1\right)^3\left(7n^2+22n+17\right)b\left(n+1\right)+q\_3\left(n+2\right)b\left(n\right)\\ &=q\_1(n+2)b(n+2)+\left(8\left(2n+1\right)^3\left(7n^2+22n+17\right)-q\_2(n+2)\right)b(n+1)\\ &=2\left(7n^2+8n+2\right)\left(2n+3\right)\left(n+2\right)^2b\left(n+2\right)-\left(7n^2+8n+2\right)\left(n+2\right)^2(n+1)b(n),\end{aligned}$$

where we have applied the recurrence relation (36), we obtain the identity for *<sup>n</sup>* <sup>þ</sup> 1, and we conclude the result. □

Sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are jointly connected as the next lemma shows. The proof is left to the reader.

**Lemma 3.3**. *For n* ≥1*, the following two identities hold*

$$
\begin{vmatrix} q\_1(n) & q\_3(n) \\ p\_1(n-1) & p\_3(n-1) \end{vmatrix} = -8Q(n)(2n-1)(2n-3)^2(n-2);\tag{41}
$$

$$
\begin{vmatrix} q\_1(n) & q\_2(n) \\ p\_1(n-1) & p\_2(n-1) \end{vmatrix} = \mathbf{1} \mathsf{6} Q(n) (2n-1) (2n-3)^3,\tag{42}
$$

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

and we conclude the proof. □

*n* � *k n*

*n* � 1 þ *k n* � 1 � �<sup>2</sup>

ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> <sup>7</sup>*n*<sup>2</sup> � <sup>20</sup>*<sup>n</sup>* <sup>þ</sup> <sup>14</sup> � �, (37)

*q*1ð Þ *n b n*ð Þ¼ *q*2ð Þ *n b n*ð Þþ � 1 *q*3ð Þ *n b n*ð Þ � 2 , *n*≥3, (36)

*<sup>q</sup>*3ð Þ *<sup>n</sup>* <sup>≔</sup> � <sup>4</sup>ð Þ *<sup>n</sup>* � <sup>2</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup> <sup>7</sup>*n*<sup>2</sup> � <sup>6</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �*:* (39)

*<sup>q</sup>*2ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>455</sup>*n*<sup>5</sup> � <sup>2427</sup>*n*<sup>4</sup> <sup>þ</sup> <sup>4850</sup>*n*<sup>3</sup> � <sup>4406</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>1728</sup>*<sup>n</sup>* � 216, (38)

In a similar way, we obtain an identity which relates numbers ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> to the

2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *b n*ð Þ� <sup>þ</sup> <sup>1</sup> *nb n*ð Þ¼ <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � �*C*<sup>2</sup>

*Proof.* We prove the identity by the induction method. For *n* ¼ 1, we directly check the identity. Suppose that the identity holds for a given number *n*. Since

> *C*2 *n*þ1

> > *C*2 *n*

� �

¼ �8*Q n*ð Þð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup>

� � � � �

where we have applied the recurrence relation (36), we obtain the identity for *<sup>n</sup>* <sup>þ</sup> 1, and we conclude the result. □ Sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are jointly connected as the next lemma

*b n*ð Þ� <sup>þ</sup> <sup>2</sup> <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � �ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup>

<sup>¼</sup> <sup>16</sup>*Q n*ð Þð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>3</sup>

, *n* ∈ *:* (35)

*<sup>n</sup>:* (40)

*b n*ð Þ þ 1

ð Þ *n* � 2 ; (41)

, (42)

ð Þ *n* þ 1 *b n*ð Þ,

Now we consider this second sequence of integer numbers defined by

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> k*¼0

*<sup>i</sup>* <sup>∈</sup>f g 1,2,3 are defined by

Note that *b*ð Þ¼ 1 1, *b*ð Þ¼ 2 3, *b*ð Þ¼ 3 19, *b*ð Þ¼ 4 163, *b*ð Þ¼ 5 1625, etc. This sequence also appears indexed in the On-Line Encyclopedia of Integer Sequences by N.J.A. Sloane [16] with the reference *A*183069, and V. Kotesovec proved the

*b n*ð Þ <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

*k*¼0

following recurrence relation:

where polynomials *qi*

square of Catalan numbers.

ð Þ *n* þ 2 *Cn*þ<sup>1</sup> ¼ 2 2ð Þ *n* þ 1 *Cn*, we have that

<sup>¼</sup> 2 7*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � �ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>3</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup>

shows. The proof is left to the reader.

� � � � �

**78**

*q*1ð Þ *n q*3ð Þ *n p*1ð Þ *n* � 1 *p*3ð Þ *n* � 1

> � � � � �

<sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � � <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>22</sup>*<sup>n</sup>* <sup>þ</sup> <sup>17</sup> � �ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup>

<sup>¼</sup> <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � � <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>22</sup>*<sup>n</sup>* <sup>þ</sup> <sup>17</sup> � �4 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

<sup>¼</sup> 4 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>22</sup>*<sup>n</sup>* <sup>þ</sup> <sup>17</sup> � �ð Þ 2 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *b n*ð Þ� <sup>þ</sup> <sup>1</sup> *nb n*ð Þ

<sup>¼</sup> *<sup>q</sup>*1ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *b n*ð Þþ <sup>þ</sup> <sup>2</sup> 8 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup> <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>22</sup>*<sup>n</sup>* <sup>þ</sup> <sup>17</sup> � � � *<sup>q</sup>*2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup>

<sup>¼</sup> 8 2ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup> <sup>7</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>22</sup>*<sup>n</sup>* <sup>þ</sup> <sup>17</sup> � �*b n*ð Þþ <sup>þ</sup> <sup>1</sup> *<sup>q</sup>*3ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *b n*ð Þ

**Lemma 3.3**. *For n* ≥1*, the following two identities hold*

*q*1ð Þ *n q*2ð Þ *n p*1ð Þ *n* � 1 *p*2ð Þ *n* � 1

� � � � �

*k n*

*Number Theory and Its Applications*

2*n* � *k* � 1 *n* � 1 � �<sup>2</sup>

� �

*<sup>q</sup>*1ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>2</sup>*n*<sup>2</sup>

**Theorem 3.2.** *For n* ≥1*, the following identity holds*

*where Q n*ð Þ <sup>≔</sup> <sup>147</sup>*n*<sup>4</sup> � <sup>546</sup>*n*<sup>3</sup> <sup>þ</sup> <sup>666</sup>*n*<sup>2</sup> � <sup>293</sup>*<sup>n</sup>* <sup>þ</sup> 34. Our last aim of this section is to show an alternative of the following identity

$$
\binom{2n}{n}^2 = \sum\_{k=0}^n \frac{3n - 2k}{n} \binom{2n - 1 - k}{n - 1}^2,\tag{43}
$$

in Theorem 3.4. An original proof is presented in ([22], Theorem 2.3 (ii)), and it is a straightforward consequence of a more general identity in combinatorial numbers ([22], Theorem 2.3 (i)). The proof which we present here allows to recognize the natural connection among the sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> and the Catalan numbers ð Þ *Cn <sup>n</sup>*≥<sup>0</sup>. Note that one may rewrite the identity (43) in an equivalent way.

**Theorem 3.4**. *For n*≥ 1*, the following identity holds*

$$((n+1)\mathbf{C}\_n)^2 = 4(\mathfrak{Z}a(n-1) - 2b(n)), \qquad n \ge 1. \tag{44}$$

*Proof.* We write by *c n*ð Þ¼ ð Þ ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> <sup>¼</sup> <sup>2</sup>*<sup>n</sup> n* � �<sup>2</sup> , and then we have to check the following identity

$$\mathcal{L}(n) = 4(\mathfrak{Z}a(n-1) - 2b(n)), \qquad n \ge 1,\tag{45}$$

where sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> are considered in the second section. Note that

$$\begin{split} &p\_1(n-1)q\_1(n)4(3a(n-1)-2b(n)) \\ &= 12q\_1(n)\left(p\_2(n-1)a(n-2) + p\_3(n-1)a(n-3)\right) \\ &- 8p\_1(n-1)\left(q\_2(n)b(n-1) + q\_3(n)b(n-2)\right) \\ &= 12a(n-2)\left(p\_1(n-1)q\_2(n) + 16Q(n)\left(2(n-1)(2n-3)^3\right)\right) \\ &+ 12a(n-3)\left(p\_1(n-1)q\_3(n) - 8Q(n)(2n-1)(2n-3)(n-2)\right) \\ &- 8p\_1(n-1)(q\_2(n)b(n-1) - 2p\_1(n-1)q\_3(n)b(n-2) \\ &= p\_1(n-1)q\_2(n)(12a(n-2) - 8b(n-1)) \\ &+ p\_1(n-1)q\_3(n)(12a(n-3) - 8b(n-2)) \\ &+ 96(2n-1)(2n-3)^2Q(n)(2(2n-3)a(n-2) - (n-2)a(n-3)), \end{split}$$

where we have applied the recurrence relations (30) and (36) and Lemma 3.3. By the induction method and Theorem 3.1, we have that

$$\begin{aligned} p\_1(n-1)q\_1(n)4(3a(n-1)-2b(n)) &= p\_1(n-1)q\_2(n)c(n-1) + p\_1(n-1)q\_3(n)c(n-2) \\ &+ 24(2n-1)(2n-3)^2Q(n)(21n-34)c(n-2) \end{aligned}$$

for *<sup>n</sup>*<sup>≥</sup> 2. Since 4 2ð Þ *<sup>n</sup>* � <sup>3</sup> <sup>2</sup> *c n*ð Þ¼ � <sup>2</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>2</sup> *c n*ð Þ � 1 for *n* ≥2, we have that 24 2ð Þ *<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>3</sup> <sup>2</sup> *Q n*ð Þð Þ 21*n* � 34 *c n*ð Þ¼ � 2 3*p*1ð Þ *n* � 1 *Q n*ð Þ*c n*ð Þ � 1 , *n*≥1*:* (46)

Finally, we get that

$$\begin{aligned} &p\_1(n-1)q\_1(n)4(3a(n-1)-2b(n)) \\ &= p\_1(n-1)c(n-1)\left(q\_2(n)+q\_3(n)\frac{\left(n-1\right)^2}{4\left(2n-3\right)^2}+3Q(n)\right), \end{aligned}$$

and

$$\begin{aligned} c(n-1) \left( q\_2(n) + q\_3(n) \frac{\left(n-1\right)^2}{4\left(2n-3\right)^2} + 3Q(n) \right) \\ = c(n-1) 8\left(7n^2 - 20n + 14\right) \left(2n-1\right)^3 \\ = c(n) n^2 2\left(7n^2 - 20n + 14\right) \left(2n-1\right) = c(n) q\_1(n), \end{aligned}$$

and we conclude the proof. □

#### **4. Moments of squares and cubes of Catalan triangle numbers**

In this section, we present some moments of squares and cubes of Catalan triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥1,*n*≥*<sup>k</sup>* <sup>≥</sup><sup>1</sup> and ð Þ *An*,*<sup>k</sup> <sup>n</sup>*≥1,*n*þ1≥*k*≥<sup>1</sup>, i.e.,

$$\sum\_{k=1}^{n} k^{m} B\_{n,k}^{j}, \qquad \sum\_{k=1}^{n+1} \left(2k-1\right)^{m} A\_{n,k}^{j}, \tag{47}$$

for *j* ¼ 2, 3 and *m* ∈ . For *m* ¼ 0, these identities are shown in [14, 24]. See a unified proof in ([22], Corollary 2.2).

Proposition 4.1. *For n* ≥1*, we have*

$$\begin{aligned} \text{i. } \sum\_{k=1}^{n} B\_{n,k}^2 &= C\_{2n-1}, \\\\ \text{ii. } \sum\_{k=1}^{n+1} A\_{n,k}^2 &= C\_{2n}, \\\\ \text{iii. } \sum\_{k=1}^{n} (-1)^k B\_{n,k}^2 &= -\frac{n+1}{2} \end{aligned}$$

*k*¼1

**Remark.** The first values of P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup> A*2 *<sup>n</sup>*,*<sup>k</sup>* are

<sup>2</sup> *Cn:*

0, 4, 32, 236, 1865, 16080, (48)

for 1≤*n*≤ 6. We are not able to find any closed formula for the general expression.

In ([13], Theorem 2), the closed expression of

$$\mathfrak{Q}\_m(n) \coloneqq \sum\_{k=1}^n k^m B\_{n,k}^2,\tag{49}$$

is given for *m* ∈ ∪ f g0 . We present now for *m* ∈f g 0, 1, ⋯, 7 . Previously, the WZ theory was used to show them in ([12], Theorem 2.1, 2.2). See also ([1], Section 5).

**Theorem 4.2**. *For n* ∈ ,

i*:* Ω0ð Þ¼ *n C*2*n*�<sup>1</sup>

Ω4ð Þ¼ *n*

Ω6ð Þ¼ *n*

<sup>Ω</sup>2ð Þ¼ *<sup>n</sup>* ð Þ <sup>3</sup>*<sup>n</sup>* � <sup>2</sup> *<sup>n</sup>*

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

<sup>4</sup>*<sup>n</sup>* � <sup>3</sup> *<sup>C</sup>*2*n*�1,

ii*:* Ω1ð Þ¼ *n* ð Þ 2*n* � 3 ð Þ *n* þ 1 *CnCn*�2, Ω3ð Þ¼ *n n*ð Þ 2*n* � 3 ð Þ *n* þ 1 *CnCn*�2, <sup>Ω</sup>5ð Þ¼ *<sup>n</sup> <sup>n</sup>* <sup>3</sup>*n*<sup>2</sup> � <sup>5</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *CnCn*�2,

<sup>Ω</sup>7ð Þ¼ *<sup>n</sup> <sup>n</sup>* <sup>6</sup>*n n*ð Þ � <sup>1</sup> <sup>2</sup> � <sup>1</sup>

*m* ∈ f g 0, 1, ⋯, 7 in the next theorem. **Theorem 4.3**. *For n*∈ *,*

<sup>Ψ</sup>2ð Þ¼ *<sup>n</sup>* �<sup>1</sup> <sup>þ</sup> <sup>4</sup>*<sup>n</sup>* <sup>þ</sup> <sup>12</sup>*n*<sup>2</sup>

4*n* � 1

ii*:* Ψ1ð Þ¼ *n* ð Þ *n* þ 1 *CnCn*�<sup>1</sup>ð Þ 4*n* � 2 , <sup>Ψ</sup>3ð Þ¼ *<sup>n</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *CnCn*�<sup>1</sup> <sup>16</sup>*n*<sup>2</sup> � <sup>2</sup> � �,

Ψ7ð Þ¼ *n* ð Þ *n* þ 1 *CnCn*�<sup>1</sup>

**Theorem 4.4**. *For n* ≥1*, we have*

<sup>2</sup> *Cnb n*ð Þ,

<sup>3</sup> � <sup>16</sup>*<sup>n</sup>* � <sup>104</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>240</sup>*n*<sup>4</sup> ð Þ 4*n* � 1 ð Þ 4*n* � 3

<sup>Ψ</sup>5ð Þ¼ *<sup>n</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *CnCn*�<sup>1</sup> <sup>96</sup>*n*<sup>3</sup> <sup>þ</sup> <sup>32</sup>*n*<sup>2</sup> � <sup>4</sup>*<sup>n</sup>* � <sup>2</sup> � �,

*<sup>n</sup>*,*<sup>k</sup>* <sup>¼</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* ð Þ <sup>2</sup>ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* <sup>2</sup> � <sup>3</sup>*a n*ð Þ � �,

i*:* Ψ0ð Þ¼ *n C*2*<sup>n</sup>*,

Ψ4ð Þ¼ *n*

i. <sup>P</sup>*<sup>n</sup> k*¼0 *B*3 *<sup>n</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>*þ<sup>1</sup>

ii. *n* P þ1 *k*¼1 *A*3

**81**

<sup>15</sup>*n*<sup>3</sup> � <sup>30</sup>*<sup>n</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>16</sup>*<sup>n</sup>* � <sup>2</sup> *<sup>n</sup>* ð Þ 4*n* � 3 ð Þ 4*n* � 5

� �ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *CnCn*�<sup>2</sup>*:*

<sup>Ψ</sup>*m*ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

*C*2*<sup>n</sup>*,

*k*¼1

*C*2*<sup>n</sup>*,

Integer sequences of numbers ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> were treated in Section 3. They play a very interesting role to describe the sums of cubes of Catalan triangle numbers, as the next result shows. See proofs and more details in ([22], Section 3).

ð Þ <sup>4</sup>*<sup>n</sup>* � <sup>1</sup> ð Þ <sup>4</sup>*<sup>n</sup>* � <sup>3</sup> ð Þ <sup>4</sup>*<sup>n</sup>* � <sup>5</sup> *<sup>C</sup>*2*<sup>n</sup>:*

<sup>1536</sup>*n*<sup>5</sup> � <sup>1536</sup>*n*<sup>4</sup> � <sup>960</sup>*n*<sup>3</sup> � <sup>160</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>20</sup>*<sup>n</sup>* <sup>þ</sup> <sup>6</sup>

<sup>2</sup>*<sup>n</sup>* � <sup>3</sup> *:*

<sup>Ψ</sup>6ð Þ¼ *<sup>n</sup>* �<sup>15</sup> <sup>þ</sup> <sup>92</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1116</sup>*n*<sup>2</sup> <sup>þ</sup> <sup>2080</sup>*n*<sup>3</sup> � <sup>4368</sup>*n*<sup>4</sup> � <sup>6720</sup>*n*<sup>5</sup> <sup>þ</sup> <sup>6720</sup>*n*<sup>6</sup>

is obtained for *m* ∈ ∪ f g0 . Now, we present the particular cases for

ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA*<sup>2</sup>

In ([13], Theorem 4, 8), the closed expression of

*C*2*n*�1,

*C*2*n*�<sup>1</sup>*:*

*<sup>n</sup>*,*<sup>k</sup>*, (50)

<sup>105</sup>*n*<sup>5</sup> � <sup>420</sup>*n*<sup>4</sup> <sup>þ</sup> <sup>588</sup>*n*<sup>3</sup> � <sup>356</sup>*<sup>n</sup>* ð Þ <sup>2</sup> <sup>þ</sup> <sup>96</sup>*<sup>n</sup>* � <sup>10</sup> *<sup>n</sup>* ð Þ 4*n* � 3 ð Þ 4*n* � 5 ð Þ 4*n* � 7

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

and

*Number Theory and Its Applications*

*c n*ð Þ � <sup>1</sup> *<sup>q</sup>*2ð Þþ *<sup>n</sup> <sup>q</sup>*3ð Þ *<sup>n</sup>* ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>2</sup>

triangle numbers ð Þ *Bn*,*<sup>k</sup> <sup>n</sup>*≥1,*n*≥*<sup>k</sup>* <sup>≥</sup><sup>1</sup> and ð Þ *An*,*<sup>k</sup> <sup>n</sup>*≥1,*n*þ1≥*k*≥<sup>1</sup>, i.e.,

*kmB <sup>j</sup>*

X*n k*¼1

unified proof in ([22], Corollary 2.2). Proposition 4.1. *For n* ≥1*, we have*

*<sup>n</sup>*,*<sup>k</sup>* ¼ *C*2*n*�1,

*<sup>n</sup>*,*<sup>k</sup>* ¼ *C*2*<sup>n</sup>*,

*<sup>n</sup>*,*<sup>k</sup>* ¼ � *<sup>n</sup>*þ<sup>1</sup>

In ([13], Theorem 2), the closed expression of

**Remark.** The first values of P*<sup>n</sup>*þ<sup>1</sup>

<sup>2</sup> *Cn:*

ð Þ �<sup>1</sup> *<sup>k</sup> B*2

i. <sup>P</sup>*<sup>n</sup> k*¼1 *B*2

ii. *n* P þ1 *k*¼1 *A*2

iii. <sup>P</sup>*<sup>n</sup> k*¼1

expression.

Section 5).

**80**

**Theorem 4.2**. *For n* ∈ ,

<sup>¼</sup> *c n*ð Þ*n*<sup>2</sup>

4 2ð Þ *<sup>n</sup>* � <sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>3</sup>*Q n*ð Þ

2 7*n*<sup>2</sup> � <sup>20</sup>*<sup>n</sup>* <sup>þ</sup> <sup>14</sup> � �ð Þ¼ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> *c n*ð Þ*q*1ð Þ *<sup>n</sup>* ,

ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup>*

*<sup>n</sup>*,*<sup>k</sup>*, (47)

*<sup>n</sup>*,*<sup>k</sup>*, (49)

!

<sup>¼</sup> *c n*ð Þ � <sup>1</sup> 8 7*n*<sup>2</sup> � <sup>20</sup>*<sup>n</sup>* <sup>þ</sup> <sup>14</sup> � �ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> <sup>3</sup>

In this section, we present some moments of squares and cubes of Catalan

*k*¼1

for *j* ¼ 2, 3 and *m* ∈ . For *m* ¼ 0, these identities are shown in [14, 24]. See a

*<sup>n</sup>*,*<sup>k</sup>*, <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

*<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

for 1≤*n*≤ 6. We are not able to find any closed formula for the general

<sup>Ω</sup>*m*ð Þ *<sup>n</sup>* <sup>≔</sup> <sup>X</sup>*<sup>n</sup>*

*k*¼1

is given for *m* ∈ ∪ f g0 . We present now for *m* ∈f g 0, 1, ⋯, 7 . Previously, the WZ theory was used to show them in ([12], Theorem 2.1, 2.2). See also ([1],

*kmB*<sup>2</sup>

*A*2 *<sup>n</sup>*,*<sup>k</sup>* are

0, 4, 32, 236, 1865, 16080, (48)

**4. Moments of squares and cubes of Catalan triangle numbers**

and we conclude the proof. □

$$\begin{aligned} \text{i. } \Omega\_0(n) &= \text{C}\_{2n-1} \\ \Omega\_2(n) &= \frac{(3n-2)n}{4n-3} \text{C}\_{2n-1}, \\ \Omega\_4(n) &= \frac{(15n^3 - 30n^2 + 16n - 2)n}{(4n-3)(4n-5)} \text{C}\_{2n-1}, \\ \Omega\_6(n) &= \frac{(105n^5 - 420n^4 + 588n^3 - 356n^2 + 96n - 10)n}{(4n-3)(4n-5)(4n-7)} \text{C}\_{2n-1}. \end{aligned}$$

$$\begin{aligned} \text{ii. } \Omega\_1(n) &= (2n-3)(n+1)C\_n C\_{n-2}, \\ \Omega\_3(n) &= n(2n-3)(n+1)C\_n C\_{n-2}, \\ \Omega\_5(n) &= n\left(3n^2 - 5n + 1\right)(n+1)C\_n C\_{n-2}, \\ \Omega\_7(n) &= n\left(6n(n-1)^2 - 1\right)(n+1)C\_n C\_{n-2}. \end{aligned}$$

In ([13], Theorem 4, 8), the closed expression of

$$\Psi\_m(n) \coloneqq \sum\_{k=1}^n (2k-1)^m A\_{n,k}^2,\tag{50}$$

is obtained for *m* ∈ ∪ f g0 . Now, we present the particular cases for *m* ∈ f g 0, 1, ⋯, 7 in the next theorem.

**Theorem 4.3**. *For n*∈ *,*

$$\begin{aligned} \text{i. } \Psi\_0(n) &= C\_{2n}, \\ \Psi\_2(n) &= \frac{-1 + 4n + 12n^2}{4n - 1} C\_{2n}, \\ \Psi\_4(n) &= \frac{3 - 16n - 104n^2 + 240n^4}{(4n - 1)(4n - 3)} C\_{2n}, \\ \Psi\_6(n) &= \frac{-15 + 92n + 1116n^2 + 2080n^3 - 4368n^4 - 6720n^5 + 6720n^6}{(4n - 1)(4n - 3)(4n - 5)} C\_{2n}. \end{aligned}$$

$$\begin{aligned} \text{ii. } \Psi\_1(n) &= (n+1) \text{C}\_n \text{C}\_{n-1} (4n-2), \\ \Psi\_3(n) &= (n+1) \text{C}\_n \text{C}\_{n-1} (16n^2 - 2), \\ \Psi\_5(n) &= (n+1) \text{C}\_n \text{C}\_{n-1} (96n^3 + 32n^2 - 4n - 2), \\ \Psi\_7(n) &= (n+1) \text{C}\_n \text{C}\_{n-1} \frac{1536n^5 - 1536n^4 - 960n^3 - 160n^2 + 20n + 6}{2n-3}. \end{aligned}$$

Integer sequences of numbers ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> were treated in Section 3. They play a very interesting role to describe the sums of cubes of Catalan triangle numbers, as the next result shows. See proofs and more details in ([22], Section 3).

**Theorem 4.4**. *For n* ≥1*, we have*

$$\begin{aligned} \text{i. } &\sum\_{k=0}^{n} B\_{n,k}^{3} = \frac{n+1}{2} \mathbf{C}\_{n} b(n), \\\\ \text{ii. } &\sum\_{k=1}^{n+1} A\_{n,k}^{3} = (n+1) \mathbf{C}\_{n} \left( \left( 2(n+1) \mathbf{C}\_{n} \right)^{2} - 3a(n) \right), \end{aligned}$$

$$\text{iii.} \sum\_{k=1}^{n+1} (-1)^k A\_{n,k}^3 = \frac{n-1}{2n+1} \binom{2n}{n} \binom{3n}{n} .$$

**Remark.** To check P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*B*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* in Theorem 4.4 (i), we need to show the identity:

$$
\binom{2n}{n}^2 = \sum\_{k=0}^n \frac{3n - 2k}{n} \binom{2n - 1 - k}{n - 1}^2, \qquad n \ge 1,\tag{51}
$$

where *Pm*�<sup>1</sup> and *Qm*�<sup>1</sup> are polynomials of integer coefficients at degree at most

**Conjecture 5.2.** After Theorem 2.1 (iii) and (iv), it is also natural to conjecture

where *Rm*�<sup>1</sup> and *Sm*�<sup>1</sup> are polynomials of integer coefficients at degree at most

*m* ∈ f g 1, 2, 3, 4 and *n*∈ f g 1, 2, 3, 4, 5 . We conjecture that the factor ð Þ *n* þ 1 *Cn* divides

**Conjecture 5.7.** The sums of alternating powers of Catalan triangle numbers *Bn*,*<sup>k</sup>*

*<sup>k</sup>*¼<sup>1</sup>*kmB*<sup>4</sup>

P*n k*¼**1** *k***3** *B***3** *n***,***k*

*n* Pþ**1** *k*¼**1**

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>3</sup>** *A***3** *n***,***k*

**Conjecture 5.3.** In **Table 1**, we present the moments P*<sup>n</sup>*

**Conjecture 5.4.** In **Table 2**, we give the moments P*<sup>n</sup>*þ<sup>1</sup>

*<sup>n</sup>*<sup>∈</sup> f g 1, 2, 3, 4, 5 in **Table 3**. Then we conjecture that the factor *<sup>n</sup>*þ<sup>1</sup>

*m* ∈ f g 1, 2, 3, 4 and *n*∈ f g 1, 2, 3, 4, 5 . We conjecture that ð Þ *n* þ 1 *Cn* divides

P*n k*¼**1** *k***2** *B***3** *n***,***k*

11 1 1 1 10 12 16 24 256 390 664 1230 8884 15, 680 30, 592 64, 400 356, 374 701, 820 1, 523, 158 3, 569, 580

**Conjecture 5.6.** In **Table 4**, we give the moments P*<sup>n</sup>*þ<sup>1</sup>

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

*n* Pþ**1** *k*¼**1**

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>2</sup>** *A***3** *n***,***k*

 4 10 28 82 94 276 862 2820 2944 9860 35, 776 139, 700 111, 010 417, 200 1, 713, 826 7, 610, 960 5 4, 677, 160 19, 342, 008 87, 730, 360 430, 535, 448

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ . **Conjecture 5.5.** We give the moments P*<sup>n</sup>*

*<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

*A*4

*k*¼**1** kB**<sup>3</sup>** *n***,***k*

*<sup>m</sup>* <sup>∈</sup> f g 1, 2, 3, 4 and *<sup>n</sup>*<sup>∈</sup> f g 1, 2, 3, 4, 5 . Then we conjecture that the factor *<sup>n</sup>*þ<sup>1</sup>

Λ2*<sup>m</sup>*ð Þ¼ *n* ð Þ *n* þ 1 *CnRm*�<sup>1</sup>ð Þ *n* , (57) <sup>Λ</sup>2*m*�<sup>1</sup>ð Þ¼ *<sup>n</sup>* <sup>2</sup><sup>2</sup>*nSm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (58)

*<sup>k</sup>*¼<sup>1</sup>*kmB*<sup>3</sup>

*<sup>k</sup>*¼<sup>1</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA*<sup>3</sup>

*<sup>n</sup>*,*<sup>k</sup>* for *m* ∈f g 1, 2, 3, 4 and

*<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA*<sup>4</sup>

*<sup>n</sup>*,*<sup>k</sup>* for

<sup>2</sup> *Cn*

*<sup>n</sup>*,*<sup>k</sup>* for

<sup>2</sup> *Cn* divides

*<sup>n</sup>*,*<sup>k</sup>* for

P*n k*¼**1** *k***<sup>4</sup>***B***<sup>3</sup>** *n***,***k*

*n* Pþ**1** *k*¼**1**

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>4</sup>***A***<sup>3</sup>**

*n***,***k*

*m* � 1.

*m* � 1.

P*<sup>n</sup>*þ<sup>1</sup>

P*<sup>n</sup>*

P*<sup>n</sup>*þ<sup>1</sup>

and *An*,*<sup>k</sup>*,

**Table 1.**

**Table 2.**

**83**

*Moments of cubes of An*,*k.*

*<sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*m*�<sup>1</sup>

divides P*<sup>n</sup>*

*<sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*mB*<sup>3</sup>

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

*<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*mA*<sup>3</sup>

*B*4

*<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*m*�<sup>1</sup>

*<sup>n</sup>* <sup>P</sup>*<sup>n</sup>*

*Moments of cubes of Bn*,*k.*

Pþ**1** *k*¼**1**

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup>** *<sup>A</sup>***<sup>3</sup>** *n***,***k*

*n n*

that for *m*, *n* ∈ ,

see ([22], Theorem 3.3). In Theorem 3.4, we have presented an alternative proof of this identity.

The first values of P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup> B*3 *<sup>n</sup>*,*<sup>k</sup>* are

$$-1, \qquad -7, \qquad -62, \qquad -215, \qquad 17332, \qquad 945342,\tag{52}$$

for 1≤*n*≤ 6. We are not able to find any closed formula for the general expression.

#### **5. Conclusions and future developments**

In this paper we have studied in detail

$$\sum\_{k=1}^{n} k^{m} B\_{n,k}^{j}, \qquad \sum\_{k=1}^{n+1} (2k-1)^{m} A\_{n,k}^{j}, \tag{53}$$

for *n*∈ and several values of *j*∈ . The main objective is to give a closed formula where a factor is *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn*, ð Þ *n* þ 1 *Cn*, *C*2*<sup>n</sup>*, or other Catalan number, for example, in Theorem 2.1, Proposition 4.1, and Theorems 4.2 and 4.3. These results complete previous studies for *m* ¼ 0, 1 and 2. In the case of *j* ¼ 3 and *m* ¼ 0, some known integer sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> appear in Theorem 4.4. Also the alternating sums

$$\sum\_{k=1}^{n}(-\mathbf{1})^{k}B\_{n,k}^{j}, \qquad \sum\_{k=1}^{n+1}(-\mathbf{1})^{k}A\_{n,k}^{j},\tag{54}$$

are considered in Theorem 2.2, Proposition 4.1 (iii), and Proposition 4.4 (iii).

To show these identities, we have combined the analytic proofs and the WZ theory which is useful to show combinatorial identities. Our results allow continuing this research, and future developments could be made.

In the following, we present some conjectures about new identities in Catalan triangle numbers. These conjectures are about the properties of divisibility of sums and alternating sums of powers of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*. The factors which we consider are *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn* and ð Þ *n* þ 1 *Cn*.

**Conjecture 5.1.** After Theorem 2.1 (i) and (ii), it is natural to conjecture that for *m*, *n* ∈

$$
\Delta\_{2m}(n) = \frac{n+1}{2} \mathcal{C}\_n n P\_{m-1}(n),
\tag{55}
$$

$$
\Delta\_{2m-1}(n) = 2^{n-m-1} Q\_{m-1}(n),
\tag{56}
$$

iii. *<sup>n</sup>* P þ1 *k*¼1

of this identity.

expression.

ð Þ �<sup>1</sup> *<sup>k</sup> A*3

*Number Theory and Its Applications*

**Remark.** To check P*<sup>n</sup>*

The first values of P*<sup>n</sup>*

formula where a factor is *<sup>n</sup>*þ<sup>1</sup>

factors which we consider are *<sup>n</sup>*þ<sup>1</sup>

alternating sums

*m*, *n* ∈

**82**

*<sup>n</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>n</sup>*�<sup>1</sup> 2*n*þ1

2*n n* � �<sup>2</sup>

2*n n* � � 3*n*

*<sup>k</sup>*¼<sup>1</sup>*B*<sup>3</sup>

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> k*¼0

*<sup>k</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

**5. Conclusions and future developments**

In this paper we have studied in detail

X*n k*¼1

> X*n k*¼1

ð Þ �<sup>1</sup> *<sup>k</sup> B j*

ing this research, and future developments could be made.

Δ2*<sup>m</sup>*ð Þ¼ *n*

*kmB <sup>j</sup>*

*n* � � *:*

3*n* � 2*k n*

> *B*3 *<sup>n</sup>*,*<sup>k</sup>* are

*<sup>n</sup>*,*<sup>k</sup>* in Theorem 4.4 (i), we need to show the identity:

, *n* ≥1, (51)

*<sup>n</sup>*,*<sup>k</sup>*, (53)

*<sup>n</sup>*,*<sup>k</sup>*, (54)

<sup>2</sup> *CnnPm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (55)

*Qm*�<sup>1</sup>ð Þ *<sup>n</sup>* , (56)

2*n* � 1 � *k n* � 1 � �<sup>2</sup>

see ([22], Theorem 3.3). In Theorem 3.4, we have presented an alternative proof

�1, � 7, � 62, � 215, 17332, 945342, (52)

for 1≤*n*≤ 6. We are not able to find any closed formula for the general

*<sup>n</sup>*,*<sup>k</sup>*, <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

*k*¼1

for *n*∈ and several values of *j*∈ . The main objective is to give a closed

example, in Theorem 2.1, Proposition 4.1, and Theorems 4.2 and 4.3. These results complete previous studies for *m* ¼ 0, 1 and 2. In the case of *j* ¼ 3 and *m* ¼ 0, some known integer sequences ð Þ *a n*ð Þ *<sup>n</sup>*≥<sup>0</sup> and ð Þ *b n*ð Þ *<sup>n</sup>*≥<sup>1</sup> appear in Theorem 4.4. Also the

*<sup>n</sup>*,*<sup>k</sup>*, <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

are considered in Theorem 2.2, Proposition 4.1 (iii), and Proposition 4.4 (iii). To show these identities, we have combined the analytic proofs and the WZ theory which is useful to show combinatorial identities. Our results allow continu-

In the following, we present some conjectures about new identities in Catalan triangle numbers. These conjectures are about the properties of divisibility of sums and alternating sums of powers of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*. The

<sup>2</sup> *Cn* and ð Þ *n* þ 1 *Cn*. **Conjecture 5.1.** After Theorem 2.1 (i) and (ii), it is natural to conjecture that for

*n* þ 1

<sup>Δ</sup>2*m*�<sup>1</sup>ð Þ¼ *<sup>n</sup>* <sup>2</sup>*<sup>n</sup>*�*m*�<sup>1</sup>

*k*¼1

ð Þ �<sup>1</sup> *<sup>k</sup> A j*

ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup>*

<sup>2</sup> *Cn*, ð Þ *n* þ 1 *Cn*, *C*2*<sup>n</sup>*, or other Catalan number, for

where *Pm*�<sup>1</sup> and *Qm*�<sup>1</sup> are polynomials of integer coefficients at degree at most *m* � 1.

**Conjecture 5.2.** After Theorem 2.1 (iii) and (iv), it is also natural to conjecture that for *m*, *n* ∈ ,

$$
\Lambda\_{2m}(n) = (n+1)\mathbf{C}\_n \mathbf{R}\_{m-1}(n),\tag{57}
$$

$$
\Lambda\_{2m-1}(n) = 2^{2n} \mathbb{S}\_{m-1}(n),
\tag{58}
$$

where *Rm*�<sup>1</sup> and *Sm*�<sup>1</sup> are polynomials of integer coefficients at degree at most *m* � 1.

**Conjecture 5.3.** In **Table 1**, we present the moments P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*kmB*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *<sup>m</sup>* <sup>∈</sup> f g 1, 2, 3, 4 and *<sup>n</sup>*<sup>∈</sup> f g 1, 2, 3, 4, 5 . Then we conjecture that the factor *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn* divides P*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*mB*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

**Conjecture 5.4.** In **Table 2**, we give the moments P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>1</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m* ∈ f g 1, 2, 3, 4 and *n*∈ f g 1, 2, 3, 4, 5 . We conjecture that the factor ð Þ *n* þ 1 *Cn* divides P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*mA*<sup>3</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

**Conjecture 5.5.** We give the moments P*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*kmB*<sup>4</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m* ∈f g 1, 2, 3, 4 and *<sup>n</sup>*<sup>∈</sup> f g 1, 2, 3, 4, 5 in **Table 3**. Then we conjecture that the factor *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn* divides P*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*k*<sup>2</sup>*m*�<sup>1</sup> *B*4 *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

**Conjecture 5.6.** In **Table 4**, we give the moments P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA*<sup>4</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m* ∈ f g 1, 2, 3, 4 and *n*∈ f g 1, 2, 3, 4, 5 . We conjecture that ð Þ *n* þ 1 *Cn* divides P*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> <sup>2</sup>*m*�<sup>1</sup> *A*4 *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n*∈ .

**Conjecture 5.7.** The sums of alternating powers of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*,


**Table 1.**

*Moments of cubes of Bn*,*k.*


**Table 2.** *Moments of cubes of An*,*k.*

$$\sum\_{k=1}^{n}(-1)^{k}B\_{n,k}^{j},\qquad\text{and}\qquad\sum\_{k=1}^{n+1}(-1)^{k}A\_{n,k}^{j},\tag{59}$$

ii. Alternating moments of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*, i.e.,

P þ1 *k*¼1

are a new interesting research which could be considered in later articles,

P þ1 *k*¼1 *bk A j*

P.J. Miana has been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS and Project E26-17R, D.G. Aragón, Spain. Natalia Romero has been partially supported by the Spanish Ministry of Science, Innovation and

In this appendix, we present some tables of powers of Catalan triangle numbers

*Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*. As we have mentioned above, they are used to conjecture some

Mathematics Subject Classification: 05A19; 05A10; 11B65, 11B75

1 Departamento de Matemáticas, Instituto Universitario de Matemáticas y

2 Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\* and Natalia Romero<sup>2</sup>

Aplicaciones, Universidad de Zaragoza, Zaragoza, Spain

\*Address all correspondence to: pjmiana@unizar.es

provided the original work is properly cited.

iii. In a similar way, weight moments of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*,

are worth studying them for some *a*, *b*∈ , compared with ([9], Theorem 1.1).

ð Þ �ð Þ <sup>2</sup>*<sup>k</sup>* � <sup>1</sup> *mA <sup>j</sup>*

*<sup>n</sup>*,*<sup>k</sup>*, (60)

*<sup>n</sup>*,*<sup>k</sup>*, *j*, *n*∈ , (61)

*<sup>n</sup>*,*<sup>k</sup>*, *<sup>n</sup>*

*<sup>n</sup>*,*<sup>k</sup>*, *<sup>n</sup>*

P*n k*¼1

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

compared with ([24], Theorem 2.3).

Universities, Project PGC2018-095896-B-C21.

**Acknowledgements**

statements in the Section 5.

**Additional information**

**Author details**

Pedro J. Miana<sup>1</sup>

Spain

**85**

**Appendix**

P*n k*¼1

*akB <sup>j</sup>*

ð Þ �*<sup>k</sup> mB <sup>j</sup>*

have been considered in this paper: in Theorem 2.2 (i) and (ii) for *j* ¼ 1, in Proposition 4.1 (iii) for *j* ¼ 2, and in Theorem 4.4 (iii) for *j* ¼ 3. In **Table 5**, we present the alternating sums of the fourth and fifth powers of Catalan triangle numbers. All these results join to conjecture that the factor *<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> *Cn* divides P*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>ð Þ �<sup>1</sup> *<sup>k</sup> B*<sup>2</sup>*<sup>m</sup> <sup>n</sup>*,*<sup>k</sup>* for *<sup>m</sup>*, *<sup>n</sup>*<sup>∈</sup> and ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* divides <sup>P</sup>*<sup>n</sup>*þ<sup>1</sup> *<sup>k</sup>*¼<sup>0</sup>ð Þ �<sup>1</sup> *<sup>k</sup> A*<sup>2</sup>*m*�<sup>1</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n* ∈ . Finally we give some general comments and ideas which could be followed in future works.


i. The generating formula (1) allows an interesting way to show some combinatorial identities in an analytic way.

#### **Table 3.**

*Moments of the fourth power of Bn*,*k.*


#### **Table 4.**

*Moments of the fourth power of An*,*k.*


#### **Table 5.**

*Sums of alternating powers of Bn*,*<sup>k</sup> and An*,*k.*

ii. Alternating moments of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*, i.e.,

$$\sum\_{k=1}^{n}(-k)^{m}B\_{n,k}^{j}, \qquad \sum\_{k=1}^{n+1}(-(2k-1))^{m}A\_{n,k}^{j},\tag{60}$$

are a new interesting research which could be considered in later articles, compared with ([24], Theorem 2.3).

iii. In a similar way, weight moments of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*,

$$\sum\_{k=1}^{n} a^k B\_{n,k}^j, \qquad \sum\_{k=1}^{n+1} b^k A\_{n,k}^j, \qquad j, n \in \mathbb{N}, \tag{61}$$

are worth studying them for some *a*, *b*∈ , compared with ([9], Theorem 1.1).

#### **Acknowledgements**

P.J. Miana has been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS and Project E26-17R, D.G. Aragón, Spain. Natalia Romero has been partially supported by the Spanish Ministry of Science, Innovation and Universities, Project PGC2018-095896-B-C21.

### **Appendix**

X*n k*¼1

*Number Theory and Its Applications*

P*<sup>n</sup>*

**Table 3.**

**Table 4.**

**Table 5.**

**84**

*n <sup>n</sup>*

*<sup>k</sup>*¼<sup>0</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

future works.

*<sup>n</sup>* <sup>P</sup>*<sup>n</sup>*

*k*¼**1** kB**<sup>4</sup>** *n***,***k*

*Moments of the fourth power of Bn*,*k.*

*Moments of the fourth power of An*,*k.*

*k*¼**1** ð Þ �**<sup>1</sup>** *<sup>k</sup> B***4** *n***,***k*

*Sums of alternating powers of Bn*,*<sup>k</sup> and An*,*k.*

*<sup>n</sup>* <sup>P</sup>*<sup>n</sup>*

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup>** *<sup>A</sup>***<sup>4</sup>** *n***,***k*

Pþ**1** *k*¼**1**

*B*<sup>2</sup>*<sup>m</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup> Bj*

numbers. All these results join to conjecture that the factor *<sup>n</sup>*þ<sup>1</sup>

combinatorial identities in an analytic way.

*n* Pþ**1** *k*¼**1**

> *n* Pþ**1** *k*¼**1** ð Þ �**<sup>1</sup>** *<sup>k</sup> A***4** *n***,***k*

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>2</sup>** *A***<sup>4</sup>** *n***,***k*

1 4 10 28 82 2 264 770 2328 7202 3 23, 440 75, 348 256, 240 925, 092 4 2, 699, 200 9, 688, 050 37, 458, 400 155, 596, 914 5 368, 708, 256 1, 458, 679, 508 6, 249, 158, 496 28, 738, 974, 308

 �1 0 �1 0 �15 64 �31 210 �370 5312 �2102 52, 800 �1295 418, 640 �7775 13, 489, 350 1, 669, 374 32, 351, 744 109, 796, 596 3, 453, 624, 720

*<sup>n</sup>*,*<sup>k</sup>* for *<sup>m</sup>*, *<sup>n</sup>*<sup>∈</sup> and ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *Cn* divides <sup>P</sup>*<sup>n</sup>*þ<sup>1</sup>

P*n k*¼**1** *k***2** *B***4** *n***,***k*

*<sup>n</sup>*,*<sup>k</sup>*, and <sup>X</sup>*<sup>n</sup>*þ<sup>1</sup>

have been considered in this paper: in Theorem 2.2 (i) and (ii) for *j* ¼ 1, in Proposition 4.1 (iii) for *j* ¼ 2, and in Theorem 4.4 (iii) for *j* ¼ 3. In **Table 5**, we present the alternating sums of the fourth and fifth powers of Catalan triangle

Finally we give some general comments and ideas which could be followed in

i. The generating formula (1) allows an interesting way to show some

11 1 1 1 18 20 24 32 1140 1658 2700 4802 119, 140 203, 760 380, 800 758, 304 15, 339, 240 29, 193, 890 60, 190, 200 132, 142, 274

*k*¼1

ð Þ �<sup>1</sup> *<sup>k</sup> A j*

*<sup>k</sup>*¼<sup>0</sup>ð Þ �<sup>1</sup> *<sup>k</sup>*

P*n k*¼**1** *k***3** *B***4** *n***,***k*

*n* Pþ**1** *k*¼**1**

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>3</sup>** *A***<sup>4</sup>** *n***,***k*

P*n k*¼**1** ð Þ �**<sup>1</sup>** *<sup>k</sup> B***5** *n***,***k*

*<sup>n</sup>*,*<sup>k</sup>*, (59)

*A*<sup>2</sup>*m*�<sup>1</sup> *<sup>n</sup>*,*<sup>k</sup>* for *m*, *n* ∈ .

P*n k*¼**1** *k***<sup>4</sup>***B***<sup>4</sup>** *n***,***k*

*n* Pþ**1** *k*¼**1**

> *n* Pþ**1** *k*¼**1** ð Þ �**<sup>1</sup>** *<sup>k</sup> A***5** *n***,***k*

ð Þ **<sup>2</sup>***<sup>k</sup>* � **<sup>1</sup> <sup>4</sup>***A***<sup>4</sup>**

*n***,***k*

<sup>2</sup> *Cn* divides

In this appendix, we present some tables of powers of Catalan triangle numbers *Bn*,*<sup>k</sup>* and *An*,*<sup>k</sup>*. As we have mentioned above, they are used to conjecture some statements in the Section 5.

### **Additional information**

Mathematics Subject Classification: 05A19; 05A10; 11B65, 11B75

#### **Author details**

Pedro J. Miana<sup>1</sup> \* and Natalia Romero<sup>2</sup>

1 Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, Zaragoza, Spain

2 Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain

\*Address all correspondence to: pjmiana@unizar.es

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Chen X, Chu W. Moments on Catalan numbers. Journal of Mathematical Analysis and Applications. 2009;**349**(2):311-316

[2] Chu W. A new combinatorial interpretation for generalized Catalan numbers. Discrete Mathematics. 1987; **65**:91-94

[3] Chu W. Summation formulae involving harmonic numbers. Filomat. 2012;**26**(1):143-152

[4] Deutsch E, Shapiro L. A survey of the Fine numbers. Discrete Mathematics. 2001;**241**:241-265

[5] Eplett WJR. A note about the Catalan triangle. Discrete Mathematics. 1979;**25**: 289-291

[6] Guo VJW, Zeng J. Factors of binomial sums from Catalan triangle. Journal of Number Theory. 2010;**130**(1): 172-186

[7] Guo VJW, Lian X. Proofs of two conjectures on Catalan triangle numbers. Journal of Difference Equations and Applications. 2018;**24**(9): 1473-1487

[8] Gutiérrez JM, Hernández MA, Miana PJ, Romero N. New identities in the Catalan triangle. Journal of Mathematical Analysis and Applications. 2008;**341**(1):52-61

[9] Lee K-H, Oh S-J. Catalan triangle numbers and binomial coefficients. Contemporary Mathematics. 2018;**713**

[10] Lang W. On polynomials related to powers of the generating function of Catalans numbers. Fibonacci Quarterly. 2000;**38**(5):408-419

[11] Hilton P, Pedersen J. Catalan numbers, their generalization and their uses. Mathematical Intelligencer. 1991; **13**:64-75

[12] Miana PJ, Romero N. Computer proofs of new identities in the Catalan triangle. In: Biblioteca de la Revista Matemática Iberoamericana. Proc. of the "Segundas Jornadas de Teoría de Números"; Madrid. 2007. pp. 203-208

Integer Sequences. 2016;**19**;

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

upenn.edu/�wilf/AeqB.html

[26] Paule P, Schneider C. Computer proofs of a new family of harmonic number identities. Advances in Applied Mathematics. 2003;**31**(2):359-378

[27] Wilf H, Zeilberger D. Rational functions certify combinatorial. Journal of the American Mathematical

[28] Benjamin AT, Quinn J. Proofs that Really Count, Dolciani Mathematical Exposition 27. Washington, DC: Mathematical Association of America;

Society. 1990;**3**:147-158

2003

**87**

[24] Zhang Z, Pang B. Several identities in the Catalan triangle. Indian Journal of Pure and Applied Mathematics. 2010;

[25] Petkovsek M, Wilf HS, Zeilberger D. *A* ¼ *B*. Wellesley: A. K. Peters Ltd.; 1997. Available from: http://www.cis.

Article 16.3.7

**41**(2):363-378

[13] Miana PJ, Romero N. Moments of combinatorial and Catalan numbers. Journal of Number Theory. 2010; **130**(8):1876-1887

[14] Shapiro LW. A Catalan triangle. Discrete Mathematics. 1976;**14**:83-90

[15] Slavík A. Identities with squares of binomial coefficients. Ars Combinatoria. 2014;**113**:377-383

[16] Sloane N. Available from: http:// www.research.att.com/

[17] Sloane N. A Handbook of Integer Sequences. New York: Academic Press; 1973

[18] Spies J. Some identities involving harmonic numbers. Mathematics of Computation. 1990;**55**:839-863

[19] Stanley RP. Enumerative Combinatorics. Vol. 2. Cambridge: Cambridge University Press; 1999

[20] Stanley RP. Catalan Numbers. Cambridge: Cambridge University Press; 2015

[21] Aigner M. Catalan-like numbers and determinants. Journal of Combinatorial Theory, Series A. 1999;**87**:33-51

[22] Miana PJ, Ohtsuka H, Romero N. Sums of powers of Catalan triangle numbers. Discrete Mathematics. 2017; **340**(10):2388-2397

[23] Brent RP, Ohtsuka H, Osborn J-AH, Prodinger H. Some binomial sums involving absolute values. Journal of

*Moments of Catalan Triangle Numbers DOI: http://dx.doi.org/10.5772/intechopen.92046*

Integer Sequences. 2016;**19**; Article 16.3.7

**References**

**65**:91-94

289-291

172-186

1473-1487

2012;**26**(1):143-152

2001;**241**:241-265

[1] Chen X, Chu W. Moments on Catalan numbers. Journal of Mathematical Analysis and

*Number Theory and Its Applications*

[12] Miana PJ, Romero N. Computer proofs of new identities in the Catalan triangle. In: Biblioteca de la Revista Matemática Iberoamericana. Proc. of the "Segundas Jornadas de Teoría de Números"; Madrid. 2007. pp. 203-208

[13] Miana PJ, Romero N. Moments of combinatorial and Catalan numbers. Journal of Number Theory. 2010;

[14] Shapiro LW. A Catalan triangle. Discrete Mathematics. 1976;**14**:83-90

[15] Slavík A. Identities with squares of

[16] Sloane N. Available from: http://

[17] Sloane N. A Handbook of Integer Sequences. New York: Academic Press;

[18] Spies J. Some identities involving harmonic numbers. Mathematics of Computation. 1990;**55**:839-863

[19] Stanley RP. Enumerative Combinatorics. Vol. 2. Cambridge: Cambridge University Press; 1999

[20] Stanley RP. Catalan Numbers. Cambridge: Cambridge University

Theory, Series A. 1999;**87**:33-51

**340**(10):2388-2397

[21] Aigner M. Catalan-like numbers and determinants. Journal of Combinatorial

[22] Miana PJ, Ohtsuka H, Romero N. Sums of powers of Catalan triangle numbers. Discrete Mathematics. 2017;

[23] Brent RP, Ohtsuka H, Osborn J-AH, Prodinger H. Some binomial sums involving absolute values. Journal of

binomial coefficients. Ars Combinatoria. 2014;**113**:377-383

www.research.att.com/

1973

Press; 2015

**130**(8):1876-1887

Applications. 2009;**349**(2):311-316

[2] Chu W. A new combinatorial interpretation for generalized Catalan numbers. Discrete Mathematics. 1987;

[3] Chu W. Summation formulae involving harmonic numbers. Filomat.

[4] Deutsch E, Shapiro L. A survey of the Fine numbers. Discrete Mathematics.

[5] Eplett WJR. A note about the Catalan triangle. Discrete Mathematics. 1979;**25**:

[6] Guo VJW, Zeng J. Factors of binomial sums from Catalan triangle. Journal of Number Theory. 2010;**130**(1):

[7] Guo VJW, Lian X. Proofs of two conjectures on Catalan triangle numbers. Journal of Difference

[8] Gutiérrez JM, Hernández MA, Miana PJ, Romero N. New identities in

[9] Lee K-H, Oh S-J. Catalan triangle numbers and binomial coefficients. Contemporary Mathematics. 2018;**713**

[10] Lang W. On polynomials related to powers of the generating function of Catalans numbers. Fibonacci Quarterly.

[11] Hilton P, Pedersen J. Catalan numbers, their generalization and their uses. Mathematical Intelligencer. 1991;

2000;**38**(5):408-419

**13**:64-75

**86**

the Catalan triangle. Journal of Mathematical Analysis and Applications. 2008;**341**(1):52-61

Equations and Applications. 2018;**24**(9):

[24] Zhang Z, Pang B. Several identities in the Catalan triangle. Indian Journal of Pure and Applied Mathematics. 2010; **41**(2):363-378

[25] Petkovsek M, Wilf HS, Zeilberger D. *A* ¼ *B*. Wellesley: A. K. Peters Ltd.; 1997. Available from: http://www.cis. upenn.edu/�wilf/AeqB.html

[26] Paule P, Schneider C. Computer proofs of a new family of harmonic number identities. Advances in Applied Mathematics. 2003;**31**(2):359-378

[27] Wilf H, Zeilberger D. Rational functions certify combinatorial. Journal of the American Mathematical Society. 1990;**3**:147-158

[28] Benjamin AT, Quinn J. Proofs that Really Count, Dolciani Mathematical Exposition 27. Washington, DC: Mathematical Association of America; 2003

Section 2

Applications

**89**
