Fermat's Last Theorem for Amateurs
Springer
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Paulo Ribenboim
Fermat's Last Theorem for Amateurs
Springer
Paulo Ribenboim Department of Mathematics and Statistics Queen's University Kingston, Ontario, K7L 3N6 Canada
With 2 Illustrations
Mathematics Subject Classification (1991): l l A x x Library of Congress Cataloging-in-Publication Data Ribenboim, Paulo. Fermat's last theorem for amateurs / Paulo Ribenboim. p. cm. Includes bibliographical references and index. ISBN 0-387-98508-5 (he. : alk. paper) 1. Fermat's last theorem. I. Title. QA244.R53 1999 512'.74—dc21 98-41246
© 1999 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 0-387-98508-5 Springer-Verlag New York Berlin Heidelberg
SPIN 10753540
Preface
It is now well known that Fermat’s last theorem has been proved. For more than three and a half centuries, mathematicians — from the great names to the clever amateurs — tried to prove Fermat’s famous statement. The approach was new and involved very sophisticated theories. Finally the long-sought proof was achieved. The arithmetic theory of elliptic curves, modular forms, Galois representations, and their deformations, developed by many mathematicians, were the tools required to complete the difficult proof. Linked with this great mathematical feat are the names of TANIYAMA, SHIMURA, FREY, SERRE, RIBET, WILES, TAYLOR. Their contributions, as well as hints of the proof, are discussed in the Epilogue. This book has not been written with the purpose of presenting the proof of Fermat’s theorem. On the contrary, it is written for amateurs, teachers, and mathematicians curious about the unfolding of the subject. I employ exclusively elementary methods (except in the Epilogue). They have only led to partial solutions but their interest goes beyond Fermat’s problem. One cannot stop admiring the results obtained with these limited techniques. Nevertheless, I warn that as far as I can see — which in fact is not much — the methods presented here will not lead to a proof of Fermat’s last theorem for all exponents.
vi
Preface
The presentation is self-contained and details are not spared, so the reading should be smooth. Most of the considerations involve ordinary rational numbers and only occasionally some algebraic (non-rational) numbers. For this reason I excluded Kummer’s important contributions, which are treated in detail in my book, Classical Theory of Algebraic Numbers and described in my 13 Lectures on Fermat’s Last Theorem (new printing, containing an Epilogue about recent results). There are already — and there will be more — books, monographs, and papers explaining the ideas and steps in the proof of Fermat’s theorem. The readers with an extended solid background will profit more from reading such writings. Others may prefer to stay with me. In summary, if you are an amateur or a young beginner, you may love what you will read here, as I made a serious effort to provide thorough and clear explanations. On the other hand, if you are a professional mathematician, you may then wonder why I have undertaken this task now that the problem has been solved. The tower of Babel did not reach the sky, but it was one of the marvels of ancient times. Here too, there are some admirable examples of ingenuity, even more remarkable considering that the arguments are strictly elementary. It would be an unforgivable error to let these gems sink into oblivion. As Jacobi said, all for “l’honneur de l’esprit humain.” August, 1997
Paulo Ribenboim
Reader
You may feel tempted to write your own (simpler) proof of Fermat’s last theorem. I have strong views about such a project. It should be written in the Constitution of States and Nations, in the Chapter of Human Rights: It is an inalienable right of each individual to produce his or her own proof of Fermat’s last theorem. However, such a solemn statement about Fermat’s last theorem (henceforth referred to as THE theorem) should be tempered by the following articles: Art. 1. No attempted proof of THE theorem should ever duplicate a previous one. Art. 2. It is a criminal offense to submit false proofs of THE theorem to professors who arduously earn their living by teaching how not to conceive false proofs of THE theorem. Infringement of the latter, leads directly to Hell. Return to Paradise only after the said criminal has understood and is able to reproduce Wiles’ proof. (Harsh punishment.)
Contents
Preface
v
Reader
vii
Acknowledgment The Problem I. Special Cases
xiii 1 3
I.1. The Pythagorean Equation
3
I.2. The Biquadratic Equation
11
I.3. Gaussian Numbers
21
I.4. The Cubic Equation
24
I.5. The Eisenstein Field
41
I.6. The Quintic Equation
49
I.7. Fermat’s Equation of Degree Seven
57
I.8. Other Special Cases
63
I.9. Appendix
71
x
Contents
II. 4 Interludes
73
II.1. p-Adic Valuations
73
II.2. Cyclotomic Polynomials
77
II.3. Factors of Binomials
79
II.4. The Resultant and Discriminant of Polynomials
95
III. Algebraic Restrictions on Hypothetical Solutions III.1. The Relations of Barlow III.2. Secondary Relations for Hypothetical Solutions IV. Germain’s Theorem
99 99 106 109
IV.1. Sophie Germain’s Theorem
109
IV.2. Wendt’s Theorem
124
IV.3. Appendix: Sophie Germain’s Primes
139
V. Interludes 5 and 6
143
V.1. p-Adic Numbers A. The Field of p-Adic Numbers B. Polynomials with p-Adic Coefficients C. Hensel’s Lemma
143 143 145 152
V.2. Linear Recurring Sequences of Second Order
156
VI. Arithmetic Restrictions on Hypothetical Solutions and on the Exponent 165 VI.1. Congruences
165
VI.2. Divisibility Conditions
184
VI.3. Abel’s Conjecture
195
VI.4. The First Case for Even Exponents
203
VII. Interludes 7 and 8
213
VII.1. Some Relevant Polynomial Identities
213
VII.2. The Cauchy Polynomials
220
Contents
VIII. Reformulations, Consequences, and Criteria
xi
235
VIII.1. Reformulation and Consequences of Fermat’s Last Theorem 235 A. Diophantine Equations Related to Fermat’s Equation 235 A1. Lebesgue A2. Christilles A3. Perrin A4. Hurwitz A5. Kapferer A6. Frey B. Reformulations of Fermat’s Last Theorem 247 VIII.2. A. B. C.
Criteria for Fermat’s Last Theorem Connection with Euler’s Totient Function Connection with the M¨obius Function Proof that a Nontrivial Solution Cannot be in Arithmetical Progression D. Criterion with a Legendre Symbol E. Criterion with a Discriminant F. Connection with a Cubic Congruence G. Criterion with a Determinant H. Connection with a Binary Quadratic Form I. The Non-Existence of Algebraic Identities Yielding Solutions of Fermat’s Equation J. Criterion with Second-Order Linear Recurrences K. Perturbation of One Exponent L. Divisibility Condition for Pythagorean Triples
253 253 255 255 256 257 263 266 267 269 270 272 273
IX. Interludes 9 and 10
277
IX.1. The Gaussian Periods
277
IX.2. Lagrange Resolvents and Jacobi Cyclotomic Function
282
X. The Local and Modular Fermat Problem
287
X.1. The Local Fermat Problem
287
X.2. Fermat Congruence
291
X.3. Hurwitz Congruence
304
X.4. Fermat’s Congruence Modulo a Prime-Power
316
xii
Contents
XI. Epilogue
359
XI.1. Attempts A. The Theorem of Kummer B. The Theorem of Wieferich C. The First Case of Fermat’s Last Theorem for Infinitely Many Prime Exponents D. The Theorem of Faltings E. The (abc) Conjecture
359 360 361
XI.2. Victory, or the Second Death of Fermat A. The Frey Curves B. Modular Forms and the Conjecture of Shimura-Taniyama C. The Work of Ribet and Wiles
366 367
XI.3. A Guide for Further Study A. Elliptic Curves, Modular Forms: Basic Texts B. Expository C. Research
375 375 375 378
XI.4. The Electronic Mail in Action
379
Appendix A. References to Wrong Proofs
381
I.
382
Papers or Books Containing Lists of Wrong Proofs
363 363 364
369 373
II. Wrong Proofs in Papers
382
III. Insufficient Attempts
387
Appendix B. General Bibliography
389
I.
The Works of Fermat
389
II. Books Primarily on Fermat
390
III. Books with References to Fermat’s Last Theorem
391
IV. Expository, Historical, and Bibliographic Papers
392
V. Critical Papers and Reviews
395
Name Index
396
Subject Index
404
Acknowledgment
Karl Dilcher has supervised the typing of this book, read carefully the text, and made many valuable suggestions. I am very grateful for his essential help. I am also indebted to various colleagues who indicated necessary corrections in early versions of this book. My thanks go especially to the late Kustaa Inkeri as well as to Takashi Agoh, Vinko Botteri, Hendrik Lenstra, Tauno Mets¨ankyl¨a, and Guy Terjanian. For the epilogue I received advice from Gerhard Frey, Fernando Gouvˆea, and Ernst Kani, to whom I express my warmest thanks.
