An Introduction to Diophantine Equations

1. Front Matter

   Pages I-XI

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2. Diophantine Equations

   1. Front Matter

      Pages 1-1

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   2. Elementary Methods for Solving Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 3-65

   3. Some Classical Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 67-116

   4. Pell-Type Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 117-145

   5. Some Advanced Methods for Solving Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 147-190

3. Solutions to Exercises and Problems

   1. Front Matter

      Pages 191-191

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   2. Solutions to Elementary Methods for Solving Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 193-263

   3. Solutions to Some Classical Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 265-287

   4. Solutions to Pell-Type Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 289-307

   5. Solutions to Some Advanced Methods in Solving Diophantine Equations

      • Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

      Pages 309-326

4. Back Matter

   Pages 327-345

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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.

• 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

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 attackDiophantine 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)

• School of Natural Sciences and Mathemati, University of Texas at Dallas, Richardson, USA

  Titu Andreescu

• Faculty of Mathematics and Computer Scie, Babeş-Bolyai University, Cluj-Napoca, Romania

  Dorin Andrica

• Faculty of Mathematics and Computer Scie, Ovidius University of Constanta, Constanta, Romania

  Ion Cucurezeanu

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