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## Meet the editor

Dr. Cheon Seoung Ryoo is a professor in Mathematics at the Hannam University. He received his PhD degree in Mathematics from the Kyushu University. Dr. Ryoo is the author of several research articles in numerical computations with guaranteed accuracy. Also, he has contributed in the field of scientific computing, p-adic functional analysis, and analytic number theory. More recently, he has been working with quantum calculus,

special functions, differential equations, and dynamical systems.

Contents

Number Theory

*by Samin Riasat*

Type Hamlet?) *by Anant P. Godbole*

Applications

*by Sudev Naduvath*

*by Jung Yoog Kang*

*by Rajesh Cherian Roy*

A New Integer-to-Integer Transform

Digit Sums and Infinite Products

Prime Numbers Distribution Line *by Shcherbakov Aleksandr Gennadiyevich*

Moments of Catalan Triangle Numbers *by Pedro J. Miana and Natalia Romero*

Modular Sumset Labelling of Graphs

and the Structure of Their Roots

**Preface XI**

**Section 1 1**

**Chapter 1 3**

**Chapter 2 27**

**Chapter 3 37**

**Chapter 4 47**

**Chapter 5 71**

**Section 2 89**

**Chapter 6 91**

**Chapter 7 107**

Determination of the Properties of (*p, q*)-Sigmoid Polynomials

The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets (or, Can a Monkey Really

## Contents



Preface

Number theory and its applications are well known for their proven properties and

This book contains a collection of selected and refereed papers on the most recent

There are so many topics related to number theory that it is hard to list them all, but

The editor of this book sought to cover various aspects of the number theory and its applications in the current project by introducing a variety of welcoming themes from the international audience of mathematicians and researchers. The editor has organized chapters as a result of research by a group of leading researchers in the number theory and related fields. The goal of this book is to provide an overview of

the current research in the field of number theory and its applications.

The main subject areas are divided into number theory and its applications.

These include "A new integer-to-integer transform", "Digit sums and infinite products", "The Borel-Cantelli lemmas, and their relationship to limit superior and limit inferior of sets", "Prime numbers distribution line", "Moments of catalan triangle numbers", "Modular sumset labelling of graphs", "Determination of the properties of (*p, q*)-sigmoid polynomials and the structure of their roots",

"I–convergence of arithmetical functions", "Identification of eigen-frequencies and mode-shapes of beams with continuous distribution of mass and elasticity and for various conditions at supports", "Some identities involving 2-variable modified degenerate Hermite polynomials arising from differential equations and distribution of their zeros", and "Elliptic curve over a local finite ring *Rn*".

excellent applicability in interdisciplinary fields of science.

potential topics include, but are not limited to, the following:

developments in number theory and its applications.

• Algebraic number theory

• Analytic number theory

• Geometric number theory

• Arithmetic combinatorics

• Arithmetic geometry

• Arithmetic topology

• Arithmetic dynamics

• Computational number theory

• Transcendental number theory

*by Abdelhakim Chillali and Lhoussain El Fadil*
