Advertisement

© 2010

An Introduction to Diophantine Equations

A Problem-Based Approach

Benefits

  • Provides reader with the main elementary methods necessary in solving Diophantine equations

  • Approaches Diophantine equations from a problem-solving standpoint, including some original exercises and solutions

  • Covers classical Diophantine equations, including linear, Pythagorean and higher degree equations, as well as exponential Diophantine equations

Textbook

Table of contents

  1. Front Matter
    Pages I-XI
  2. Diophantine Equations

    1. Front Matter
      Pages 1-1
    2. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 3-65
    3. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 67-116
    4. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 117-145
    5. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 147-190
  3. Solutions to Exercises and Problems

    1. Front Matter
      Pages 191-191
    2. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 193-263
    3. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 265-287
    4. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 289-307
    5. Titu Andreescu, Dorin Andrica, Ion Cucurezeanu
      Pages 309-326
  4. Back Matter
    Pages 327-345

About this book

Introduction

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 material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method, inequalities, the parametric method, modular arithmetic, mathematical induction, Fermat's method of infinite descent, and the method of quadratic fields; Part II contains complete solutions to all exercises in Part I. 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.

Keywords

Fermat's method of infinite descent Gaussian integers Pell-type equations Pythagorean triples arithmetic boundary element method class diophantine equation equation history of mathematics mathematics number theory parametric method presentation techniques

Authors and affiliations

  1. 1.School of Natural Sciences and MathematiUniversity of Texas at DallasRichardsonUSA
  2. 2.Faculty of Mathematics and Computer ScieBabeş-Bolyai UniversityCluj-NapocaRomania
  3. 3.Faculty of Mathematics and Computer ScieOvidius University of ConstantaConstantaRomania

Bibliographic information

  • Book Title An Introduction to Diophantine Equations
  • Book Subtitle A Problem-Based Approach
  • Authors Titu Andreescu
    Dorin Andrica
    Ion Cucurezeanu
  • DOI https://doi.org/10.1007/978-0-8176-4549-6
  • Copyright Information Springer Science+Business Media, LLC 2010
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-0-8176-4548-9
  • eBook ISBN 978-0-8176-4549-6
  • Edition Number 1
  • Number of Pages XI, 345
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    Algebra
  • Buy this book on publisher's site
Industry Sectors
Aerospace
Electronics
Finance, Business & Banking

Reviews

From the reviews:

“This book is devoted to problems from mathematical competitions involving diophantine equations. … Each chapter contains a large number of solved examples and presents the reader with problems whose solutions can be found in the book’s second part. This volume will be particularly interesting for participants in mathematical contests and their coaches. It will also give a lot of pleasure to everyone who likes to tackle elementary, yet nontrivial problems concerning diophantine equations.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012)

“This book explains methods for solving problems with Diophantine equations that often appear in mathematical competitions at various levels. … The book can be recommended to mathematical contest participants, but also to undergraduate students, advanced high school students and teachers.” (Andrej Dujella, Mathematical Reviews, Issue 2011 j)

“Diophantus’ Arithmetica is a collection of problems each followed by a solution...The book at hand is intended for high school students, undergraduates and math teachers. It is written in a language that everyone in these groups will be familiar with. The exposition is very lucid and the proofs are clear and instructive. The book will be an invaluable source for math contest participants and other math fans. It will be an excellent addition to any math library.” (Alex Bogomolny, The Mathematical Association of America, October, 2010)

“Diophantine analysis, the business of solving equations with integers, constitutes a subdiscipline within the larger field of number theory. One problem in this subject, Fermat's last theorem, till solved, topped most lists of the world's most celebrated unsolved mathematics problems, so the subject attracted much attention from mathematicians and even the larger public. Nevertheless, sophisticated 20th-century tools invented to attack Diophantine equations (algebraic number fields, automorphic forms, L-functions, adelic groups, etc.) have emerged as proper objects of study in their own right. So for a popular subject, modern lower-level works focused on the individual Diophantine equation (and not on big machines aimed generally at classes of such equations) are relatively rare. The present volume…fills this need...Summing Up: Recommended. Lower- and upper-division undergraduates and general readers.” (D.V. Feldman, Choice, July, 2010)