Titu Andreescu Dorin Andrica Ion Cucurezeanu AnE Introduction to Diophantine Equations A Problem-Based Approach Titu Andreescu Dorin Andrica School of Natural Sciences Faculty of Mathematics and Mathem atics and Computer Science University of Texas at Dallas Babeş -Bolyai University 800 W. Campbell Road Str. Kogalniceanu 1 Richardson, TX 75080, USA 3400 Cluj-Napoca, Romania [email protected] [email protected] and Ion Cucurezeanu King Saud University Faculty of Mathematics Department of Mathematics and Computer Science College of Science Ovidius University of Constanta Riyadh 11451, Saudi Arabia B-dul Mamaia, 124 [email protected] 900527 Constanta, Romania ISBN 978-0-8176-4548-9 e-ISBN 978-0-8176–4549-6 DOI 10.1007/978-0-8176-4549-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934856 Mathematics Subject Classification (2010): 11D04, 11D09, 11D25, 11D41, 11D45, 11D61, 11D68, 11-06, 97U40 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer, software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser is part of Springer Science+Business Media www.birkhauser-science.com Preface Diophantus, the “father of algebra,” is best known for his book Arith- metica, a work on the solution of algebraic equations and the theory of numbers. However, essentially nothing is known of his life, and there has been much debate regarding precisely the years in which he lived. Diophantus did his work in the great city of Alexandria. At this time, Alexandria was the center of mathematical learning. The period from 250 bce to 350 ce in Alexandria is known as the Silver Age, also the Later Alexandrian Age. This was a time when mathe- maticians were discovering many ideas that led to our current con- ception of mathematics. The era is considered silver because it came after the Golden Age, a time of great development in the field of mathematics. This Golden Age encompasses the lifetime of Euclid. vi Preface The quality of mathematics from this period was an inspiration for the axiomatic methods of today’s mathematics. While it is known that Diophantus lived in the Silver Age, it is hard to pinpoint the exact years in which he lived. While many refer- ences to the work of Diophantus have been made, Diophantus himself made few references to other mathematicians’ work, thus making the process of determining the time that he lived more difficult. Diophantus did quote the definition of a polygonal number from the work of Hypsicles, who was active before 150 bce,sowecan conclude that Diophantus lived after that date. From the other end, Theon, a mathematician also from Alexandria, quoted the work of Diophantus in 350 ce. Most historians believe that Diophantus did most of his work around 250 ce. The greatest amount of information about Diophantus’s life comes from the possibly fictitious collection of riddles written by Metrodorus around 500 ce.Oneoftheseisas follows: His boyhood lasted 1/6 of his life; he married after 1/7 more; his beard grew after 1/12 more, and his son was born five years later; the son lived to half his father’s age, and the father died four years after the son. Diophantus was the first to employ symbols in Greek algebra. He used a symbol (arithmos) for an unknown quantity, as well as symbols for algebraic operations and for powers. Arithmetica is also significant for its results in the theory of numbers, such as the fact that no integer of the form 8n + 7 can be written as the sum of three squares. Preface vii Arithmetica is a collection of 150 problems that give approximate solutions to equations up to degree three. Arithmetica also contains equations that deal with indeterminate equations. These equations deal with the theory of numbers. The original Arithmetica is believed to have comprised 13 books, but the surviving Greek manuscripts contain only six. The others are considered lost works. It is possible that these books were lost in a fire that occurred not long after Diophantus finished Arithmetica. In what follows, we call a Diophantine equation an equation of the form f(x1,x2,...,xn)=0, (1) where f is an n-variable function with n ≥ 2. If f is a polynomial with integral coefficients, then (1) is an algebraic Diophantine equation. 0 0 0 n An n-uple (x1,x2,...,xn) ∈ Z satisfying (1) is called a solution to equation (1). An equation having one or more solutions is called solvable. Concerning a Diophantine equation three basic problems arise: Problem 1. Is the equation solvable? Problem 2. If it is solvable, is the number of its solutions finite or infinite? Problem 3. If it is solvable, determine all of its solutions. Diophantus’s work on equations of type (1) was continued by Chinese mathematicians (third century), Arabs (eight through twelfth centuries) and taken to a deeper level by Fermat, Euler, viii Preface Lagrange, Gauss, and many others. This topic remains of great importance in contemporary mathematics. This book is organized in two parts. The first contains three chapters. Chapter 1 introduces the reader to the main elementary methods in solving Diophantine equations, such as decomposition, modular arithmetic, mathematical induction, and Fermat’s infinite descent. Chapter 2 presents classical Diophantine equations, includ- ing linear, Pythagorean, higher-degree, and exponential equations, such as Catalan’s. Chapter 3 focuses on Pell-type equations, serving again as an introduction to this special class of quadratic Diophan- tine equations. Chapter 4 contains some advanced methods involv- ing Gaussian integers, quadratic rings, divisors of certain forms, and quadratic reciprocity. Throughout Part I, each of the sections con- tains representative examples that illustrate the theory. Part II contains complete solutions to all exercises in Part I. For several problems, multiple solutions are presented, along with useful comments and remarks. Many of the selected exercises and problems are original or have been given original solutions by the authors. The book is intended for undergraduates, high school students and teachers, mathematical contest (including Olympiad and Putnam) participants, as well as any person interested in mathematics. We would like to thank Richard Stong for his careful reading of the manuscript. His pertinent suggestions have been very useful in improving the text. June 2010 Titu Andreescu Dorin Andrica Ion Cucuruzeanu Contents Preface v I Diophantine Equations 1 I.1 Elementary Methods for Solving Diophantine Equations 3 1.1TheFactoringMethod................ 3 1.2 Solving Diophantine Equations Using Inequalities . 13 1.3TheParametricMethod................ 20 1.4TheModularArithmeticMethod.......... 29 1.5 The Method of Mathematical Induction . 36 1.6 Fermat’s Method of Infinite Descent (FMID) . 47 1.7 Miscellaneous Diophantine Equations . 58 x Contents I.2 Some Classical Diophantine Equations 67 2.1LinearDiophantineEquations............ 67 2.2 Pythagorean Triples and Related Problems . 76 2.3OtherRemarkableEquations............. 88 I.3 Pell-Type Equations 117 3.1 Pell’s Equation: History and Motivation . 118 3.2 Solving Pell’s Equation . .........121 3.3 The Equation ax2 − by2 =1 .............135 3.4 The Negative Pell’s Equation . .........140 I.4 Some Advanced Methods for Solving Diophantine Equations 147 4.1 The Ring Z[i]ofGaussianIntegers..........151 √ 4.2 The Ring of Integers of Q[ d] ............162 4.3 Quadratic Reciprocity and Diophantine Equations . 178 4.4DivisorsofCertainForms...............181 4.4.1 Divisors of a2 + b2 ...............182 4.4.2 Divisors of a2 +2b2 ..............186 4.4.3 Divisors of a2 − 2b2 ..............188 II Solutions to Exercises and Problems 191 II.1 Solutions to Elementary Methods for Solving Diophantine Equations 193 1.1TheFactoringMethod................193 1.2 Solving Diophantine Equations Using Inequalities . 202 1.3TheParametricMethod................213 Contents xi 1.4TheModularArithmeticMethod..........219 1.5 The Method of Mathematical Induction . 229 1.6 Fermat’s Method of Infinite Descent (FMID) . 239 1.7 Miscellaneous Diophantine Equations . 253 II.2 Solutions to Some Classical Diophantine Equations 265 2.1LinearDiophantineEquations............265 2.2 Pythagorean Triples and Related Problems . 273 2.3OtherRemarkableEquations.............278 II.3 Solutions to Pell-Type Equations 289 3.1 Solving Pell’s Equation by Elementary Methods . 289 3.2 The Equation ax2 − by2 =1 .............298 3.3 The Negative Pell’s Equation . .........301 II.4 Solutions to Some Advanced Methods in Solving Diophantine Equations 309 4.1 The Ring Z[i]ofGaussianIntegers..........309 √ 4.2 The Ring of Integers of Q[ d] ............314 4.3 Quadratic Reciprocity and Diophantine Equations . 322 4.4DivisorsofCertainForms...............324 References 327 Glossary 331 Index 341 Part I Diophantine Equations I.1 Elementary Methods for Solving Diophantine Equations 1.1 The Factoring Method Given the equation f(x1,x2,...,xn) = 0, we write it in the equiva- lent form f1(x1,x2,...,xn)f2(x1,x2,...,xn) ···fk(x1,x2,...,xn)=a, where f1,f2,...,fk ∈ Z[X1,X2,...,Xn]anda ∈ Z.Giventheprime factorization of a, we obtain finitely many decompositions into k integer factors a1,a2,...,ak. Each such factorization yields a system of equations ⎧ ⎪ ⎪ f1(x1,x2,...,xn)=a1, ⎪ ⎨⎪ f2(x1,x2,...,xn)=a2, ⎪ . ⎪ . ⎪ ⎩⎪ fk(x1,x2,...,xn)=ak. Solving all such systems gives the complete set of solutions to (1).