An Introduction to Diophantine Equations

As one can see from the title, this book is an introduction to the study of diophantine equations. The material is organized in two parts. The first part contains three chapters. Chapter 1 introduces the reader to the main elementary methods in solving diophantine equations such as decomposition, modular arithmetic, mathematical induction, Fermat's infinite descent. Chapter 2 presents some classical diophantine equations, including linear, pythagorean and some higher degree equations. Chapter 3 focuses on Pell's-type equations, serving again as an introduction to this special class of quadratic diophantine equations. Throughout Part I, each of the sections contains representative examples that illustrate the theoretical part.

Part II contains the complete solutions to all exercises featured in Part I. For several problems multiple solutions are included, along with useful comments and remarks. Many of the selected exercises and problems are original or have been give original solutions.

The book is intended for undergraduates, high school students and their teachers, mathematical contest (including Olympiad and Putnam) participants, as well as any person interested in essential mathematics.

Preface

Part 1. Diophantine Equations
Chapter 1. Elementary Methods for Solving Diophantine Equations
1.1. The Decomposition Method
1.2. Solving Diophantine Equations Using Inequalities
1.3. The Parametric Method
1.4. The Modular Arithmetic Method
1.5. The Method of Mathematical Induction
1.6 .Fermat's Method of Infinite Descent (FMID)
1.7. Miscellaneous Diophantine Equations

Chapter 2. Some Classical Diophantine Equations
2.1. Linear Diophantine Equations
2.2. Pythagorean Triples and Related Problems
2.3. Other Remarkable Equations

Chapter 3. Pell's-Type Equations
3.1. Pell's Equation: History and Motivation
3.2. Solving Pell's Equation by Elementary Methods
3.3. The Equation ax2-by2=1
3.4. The Negative Pell's Equation

Part 2. Solutions to Exercises and Problems
Chapter 1. Elementary Methods for Solving Diophantine Equations
1.1. The Decomposition Method
1.2. Solving Diophantine Equations Using Inequalities
1.3. The Parametric Method
1.4. The Modular Arithmetic Method
1.5. The Method of Mathematical Induction
1.6 .Fermat's Method of Infinite Descent (FMID)
1.7. Miscellaneous Diophantine Equations

Chapter 2. Some Classical Diophantine Equations
2.1. Linear Diophantine Equations
2.2. Pythagorean Triples and Related Problems
2.3. Other Remarkable Equations

Chapter 3. Pell's-Type Equations
3.1. Pell's Equation: History and Motivation
3.2. Solving Pell's Equation by Elementary Methods
3.3. The Equation ax2-by2=1
3.4. The Negative Pell's Equation

Bibliography
Index