Number theory is a branch of pure mathematics that deals with the study of natural numbers and integers. It focuses on the set of positive whole numbers which are called the set of natural numbers. The number theory helps in discovering interesting relationships between different sorts of numbers and in proving their legitimacy. It includes both experimental and theoretical aspects. The experimental aspect generates new questions and discovers their answer. The theoretical aspect tries to devise an argument that provides a conclusive answer to the questions. This book covers in detail some existent theories and innovative concepts revolving around number theory. Most of the topics introduced herein cover new techniques and applications of number theory. This book is appropriate for students seeking detailed information in this area as well as for experts. This book effectively builds a bridge from basic number theory to recent advances in applied number theory. Number theory is a branch of pure mathematics that deals with the study of natural numbers and integers. It focuses on the set of positive whole numbers which are called the set of natural numbers. The number theory helps in discovering interesting relationships between different sorts of numbers and in proving their legitimacy. It includes both experimental and theoretical aspects. The experimental aspect generates new questions and discovers their answer. The theoretical aspect tries to devise an argument that provides a conclusive answer to the questions. This book covers in detail some existent theories and innovative concepts revolving around number theory. Most of the topics introduced herein cover new techniques and applications of number theory. This book is appropriate for students seeking detailed information in this area as well as for experts. This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory.
Vahagn Aslanyan is a Dame Kathleen Ollerenshaw Fellow in the Department of Mathematics at the University of Manchester, also supported by an EPSRC Open Fellowship. From 2022 to 2023 he was a Leverhulme Trust Early Career Fellow at the University of Leeds. From 2019 to 2022 he was a senior research associate at the University of East Anglia. From 2017 to 2019 he was a postdoctoral associate at Carnegie Mellon University and concurrently a junior researcher at the National Academy of Sciences of Armenia. He did PhD (DPhil) at the University of Oxford from 2013 to 2017. He received bachelor's degree from Yerevan State University in 2013. His primary research interests are in Number Theory model, a branch of mathematical logic with a broad range of applications. More specifically, he is interested in applications of model theory in number theory, geometry, algebra, and analysis.
| Preface Chapter 1. Introduction to Number Theory History of Number Theory Integers and Basic Properties Prime Numbers Divisibility Rules Euclidean Algorithm Chapter 2. Modular Arithmetic and Congruences Modular Arithmetic Properties of Congruences Modular Addition and Multiplication Linear Congruences Residue Classes Chapter 3. Diophantine Equations Linear Diophantine Equations Integer Solutions Pythagorean Triples Pell’s Equation Hilbert’s Tenth Problem Chapter 4. Arithmetic Functions and Multiplicativity Divisor Function Möbius Function Multiplicative Functions Dirichlet Convolution Chapter 5. Prime Numbers and Primality Testing Prime-Counting Function Sieve of Eratosthenes Bertrand’s Postulate Twin Primes Aks Primality Test |
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