Andreescu / Andrica / Cucurezeanu An Introduction to Diophantine Equations

A Problem-Based Approach
1. Auflage 2010
ISBN: 978-0-8176-4549-6
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 350 Seiten

This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants - including Olympiad and Putnam competitors - as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.

Autoren/Hrsg.

Andreescu, Titu

Andrica, Dorin

Cucurezeanu, Ion

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;Part I Diophantine Equations;14
3.1;I.1 Elementary Methods for Solving Diophantine Equations;15
3.1.1;1.1 The Factoring Method;15
3.1.2;1.2 Solving Diophantine Equations Using Inequalities;25
3.1.3;1.3 The Parametric Method;32
3.1.4;1.4 The Modular Arithmetic Method;41
3.1.5;1.5 The Method of Mathematical Induction;48
3.1.6;1.6 Fermat’s Method of Infinite Descent (FMID);59
3.1.7;1.7 Miscellaneous Diophantine Equations;70
3.2;I.2 Some Classical Diophantine Equations;78
3.2.1;2.1 Linear Diophantine Equations;78
3.2.2;2.2 Pythagorean Triples and Related Problems;87
3.2.3;2.3 Other Remarkable Equations;99
3.2.3.1;2.3.1. Some Quadratic Diophantine Equations and Related Problems;99
3.2.3.2;2.3.2. Some Higher-Degree Diophantine Equations;111
3.3;I.3 Pell-Type Equations;128
3.3.1;3.1 Pell’s Equation: History and Motivation;129
3.3.2;3.2 Solving Pell’s Equation;132
3.3.3;3.3 The Equation ax2 by2 = 1;146
3.3.4;3.4 The Negative Pell’s Equation;151
3.4;I.4 Some Advanced Methods for Solving Diophantine Equations;157
3.4.1;4.1 The Ring Z[i] of Gaussian Integers;161
3.4.2;4.2 The Ring of Integers of Q[vd];172
3.4.3;4.3 Quadratic Reciprocity and Diophantine Equations;188
3.4.4;4.4 Divisors of Certain Forms;191
3.4.4.1;4.4.1 Divisors of a2 +b2;192
3.4.4.2;4.4.2 Divisors of a2 +2b2;196
3.4.4.3;4.4.3 Divisors of a2 2b2;198
4;Part II Solutions to Exercises and Problems;201
4.1;II.1 Solutions to Elementary Methods for Solving Diophantine Equations;202
4.1.1;1.1 The Factoring Method;202
4.1.2;1.2 Solving Diophantine Equations Using Inequalities;211
4.1.3;1.3 The Parametric Method;222
4.1.4;1.4 The Modular Arithmetic Method;228
4.1.5;1.5 The Method of Mathematical Induction;238
4.1.6;1.6 Fermat’s Method of Infinite Descent (FMID);248
4.1.7;1.7 Miscellaneous Diophantine Equations;262
4.2;II.2 Solutions to Some Classical Diophantine Equations;273
4.2.1;2.1 Linear Diophantine Equations;273
4.2.2;2.2 Pythagorean Triples and Related Problems;281
4.2.3;2.3 Other Remarkable Equations;286
4.3;II.3 Solutions to Pell-Type Equations;296
4.3.1;3.1 Solving Pell’s Equation by Elementary Methods;296
4.3.2;3.2 The Equation ax2 by2 = 1;305
4.4;II.4 Solutions to Some Advanced Methods in Solving Diophantine Equations;315
4.4.1;4.1 The Ring Z[i] of Gaussian Integers;315
4.4.2;4.2 The Ring of Integers of Q[vd];320
4.4.3;4.3 Quadratic Reciprocity and Diophantine Equations;328
4.4.4;4.4 Divisors of Certain Forms;330
5;References;333
6;Glossary;337
7;Index;346