**1. Introduction**

No algorithm that produces prime numbers in explicit forms, or rather, this goal was not reached, mathematicians resorted to an alternative method to discover prime numbers, which are primitive tests since Fermat's era or before, and this method proved its effectiveness to the extent that many prime numbers were discovered The Great Until. Euler studied Fermat's prime numbers and discovered some of them. Cullen, Broth and Mersinne also studied those numbers, as well as Pedro Berrizbeitia, Wieb Bosma and A. Schönhage. The results that we reached in this study are in the same way as those who followed the work of [1–3]. In a paper published on March 11, 2011 MO [3] prove the following result. *Cm*ð Þ *<sup>a</sup>* is a prime where *Cm*ð Þ¼ *<sup>a</sup> mam* <sup>þ</sup> 1 then

*ma<sup>m</sup>* � �ð Þ<sup>1</sup> *<sup>a</sup>* ð Þ *mod Cm*ð Þ *a* And in a paper published on July 10, 4102 using the same ideas found in MO [2], they proved [3] the following result. Let *<sup>N</sup>* <sup>¼</sup> *kpn* <sup>þ</sup> <sup>1</sup>*:*

where is p is odd prime and *k < p<sup>n</sup>* Assume that *a*∈ is a p-th power non-residue modulo *N*, then *N* is a prime if only if *ϕ<sup>p</sup> a N*�1 *p* � � � <sup>0</sup>ð Þ *mod N* . The numbers in form *Cm*ð Þ¼ *<sup>a</sup> ma<sup>m</sup>* <sup>þ</sup> 1 are called Cullen numbers, first studied by Cullen in 1905. And the numbers in the form of *kp<sup>n</sup>* <sup>þ</sup> 1 are called the Broth numbers and we call the number primes the form *MP* <sup>¼</sup> <sup>2</sup>*<sup>P</sup>* � 1 mersenne number discovered in 2005 by Martin nowak the largest prime number of Mersenne *M*<sup>25964951</sup> and 42 in the list. We know about Mersenne's number if *Mp* it is not prime then there is a prime number *q* ¼ 2*pr* þ 1 where *Mp=q* example *M*<sup>11</sup> of a non-prime. Also there is a relationship between Mersenne prime and the perfect numbers. And number in form *Fn* <sup>¼</sup> <sup>2</sup><sup>2</sup>*<sup>n</sup>* þ 1 are called Fermat numbers were first studied by Pierre de Fermat, The importance of these numbers lies in providing the large prime numbers of the known. All the large prime numbers are in the form *man* <sup>þ</sup> *<sup>b</sup>* or *<sup>a</sup><sup>n</sup>* <sup>þ</sup> *<sup>b</sup>* , for example, in 2021,

<sup>2525532</sup>*:*732525532 <sup>þ</sup> 1 was discovered the largest prime number defined by Tom Greer. There is a program in the Internet called [Prime Grid] The goal of discovering this is a kind of numbers See ½ � *https* : *∕∕primegrid:com* Researchers use several techniques in the study such as preliminary tests and high-precision computers. Prove Broth if *N* ¼ *<sup>k</sup>*2*<sup>n</sup>* <sup>þ</sup> 1 where K is odd and *<sup>k</sup> <sup>&</sup>lt;*2*<sup>n</sup>* if *<sup>a</sup> N*�1 <sup>2</sup> � �1ð Þ *mod N* same *a*∈ then N is a prime. The next important step was made in 1914 by Pocklington his result is the first generalization of Proth's theorem suitable for numbers of the form prove

Pocklington if *<sup>N</sup>* <sup>¼</sup> *Kpn* <sup>þ</sup> 1 where K is odd and *<sup>k</sup> <sup>&</sup>lt;pn* if for same *<sup>a</sup>*∈ *<sup>a</sup><sup>N</sup>*�<sup>1</sup> �

�1ð Þ *mod N and g:c:d aN*�<sup>1</sup> *<sup>p</sup>* � 1, *p* � � then N is prime. There are many works that discuss Broth's theorem and numbers. Case *p* ¼ 3 studied by W. Bosma [4] and A. Guthmann [5] Also, for a discussion on the Broth numbers, see H.C Williams [6, 7] P. Berrizbeitia [8, 9].

The purpose of this work is to study the numbers in model *mam* <sup>þ</sup> *bm* <sup>þ</sup> 1 and *<sup>p</sup>* <sup>¼</sup> *ba* þ 1 where *a*, *b >*1∈ and *p* ¼ *aq* þ 1, *a*, *q* ∈ where q is an odd prime number, In addition, tests for Fermat and Mersenne numbers are presented and the study of the relationship between two prime numbers and a polynomial with finite properties. From the results we obtained we proved, for example, *if p* and *q* are prime numbers,

*<sup>p</sup>* <sup>¼</sup> *qa* <sup>þ</sup> 1 where *<sup>q</sup>*, *<sup>a</sup> <sup>&</sup>gt;*1<sup>∈</sup> and where *Cm*ð Þ¼ *<sup>a</sup> mam* <sup>þ</sup> 1 and *<sup>π</sup><sup>j</sup>* <sup>¼</sup> <sup>1</sup> *q q j* � � then

$$\sum\_{j=1}^{q-2} \pi\_j (-\mathcal{C}\_m(a))^{q-j-1} (q - a^m) \equiv \chi\_{(m, q - a^m)}(mod \ p) \tag{1}$$

Our approach to the proof differs from the one in [2, 3]. We explicitly relied on the binomial theorem, elementary algebra, and Fermat's litter theorem. A deductive method of analysis using basic operations in elementary algebra.
