Number Theory

Challenge your problem-solving aptitude in number theory with powerful problems that have concrete examples which reflect the potential and impact of theoretical results. Each chapter focuses on a fundamental concept or result, reinforced by each of the subsections, with scores of challenging problems that allow you to comprehend number theory like never before. All students and coaches wishing to excel in math competitions will benefit from this book as will mathematicians and adults who enjoy interesting mathematics.

Based on lectures given by the authors at the AwesomeMath Summer Program over several years, this book is a slightly non-standard introduction to elementary number theory. Nevertheless, it still develops theoretical concepts from scratch with full proofs. The book insists on exemplifying these results through interesting and rather challenging problems. In particular, the reader will not find many advanced concepts in this book, but will encounter quite a lot of intriguing results that can be proven using “basic” number theory yet nonetheless test one’s problem-solving aptitude.

The book is divided into six large chapters, each focusing on a fundamental concept or result. Each chapter is itself divided into sections that reinforce a specific topic through a large series of examples arranged (subjectively) in increasing order of difficulty.

Contents

1. Introduction
2. Divisibility
3. GCD and LCM
4. The fundamental theorem of arithmetic
5. Congruences involving prime numbers
6. p-adic valuations and the distribution of primes
7. Congruences for composite moduli
8. Solutions to practice problems