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An Introduction to Diophantine Equations: A Problem-Based Approach

Titu Andreescu, Dorin Andrica and Ion Cucurezeanu
Publisher: 
Birkhäuser
Publication Date: 
2010
Number of Pages: 
345
Format: 
Hardcover
Price: 
59.95
ISBN: 
9780817645489
Category: 
Textbook
[Reviewed by
Alex Bogomolny
, on
10/28/2010
]

Diophantus’ Arithmetica is a collection of problems each followed by a solution. My first thought on picking up a book with the subtitle “A Problem-Based Approach” was that this is quite an appropriate way to treat Diophantine equations: follow in the footsteps of Diophantus himself. But the book actually goes beyond the classic format. The statements in the book are of three sorts: theorems, examples, and exercises and problems. All come with solutions (proofs for the theorems) in the spirit of Diophantus, but the distinction makes it easier for a student to grasp the essential results, followed up by the first applications and then the means to verify his/her understanding of the material. The book also contains some historical information on Diophantus himself (in the Preface), occasional remarks, while, in case of Pell’s equation, a whole section in the third chapter offers a historic sketch of the treatment of quadratic equations in integers.

The book consists of two parts. The authors claim (viii) that there are three chapters in the first part. In fact there are four. This is the most amusing typo in a book on solving equations in integers. Chapters are subdivided into topical sections. At the end of each section there are numerous problems whose solutions are collected in Part II of the book that, naturally enough, also consists of four chapters. Many examples and exercises come from various national and international olympiads; some are original (as are many solutions) with the authors.

Diophantus’ preface to his Arithmetica is very suitable for the book at hand:

Now let us tread the path to the propositions themselves, which contain a great mass of material comprised into various species. As they are both numerous and very complex to express, they are only slowly grasped by those into whose hands they are put, and include things hard to remember; for this reason I have tried to divide them up according to their subject-matter, and especially to place, as is fitting, the elementary propositions at the beginning in order that passage may be made from the simpler to the more complex. (cited from I. Thomas, Greek Mathematical Works I)

Chapter 1 is introductory where the authors highlight several elementary methods for solving Diophantine equations. The main among these are the factoring, the parametric method, modular arithmetic, induction, and infinite descent. Chapter 2 deals with the classical equations: linear and Pythagorean in the first two sections, with additional examples in the third section. Chapter 3 is devoted to Pell’s equation. Chapter 4 covers advanced methods, like Gaussian integers, rings Q[d], quadratic reciprocity.

Diophantus is deservedly considered the “father of algebra” and is so referred to in the book. Millennia have passed since Diophantus pioneered special notations for mathematical objects. He may have been understandably apprehensive whether his symbolism for the propositions that were “very complex to express” would be well received.

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 is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math.

Preface.-Part 1: Diophantine Equations.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat’s Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell’s-Type Equations.-Pell’s Equation: History and Motivation.-Solving Pell’s Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell’s Equation.-Part 2: Solutions to Exercises and Problems.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat’s Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell’s-Type Equations.-Solving Pell’s Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell’s Equation.-References.-Index.

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