# Fermat’s Last Theorem for Amateurs

Springer Science & Business Media, Mar 10, 2000 - Mathematics - 408 pages
ItisnowwellknownthatFermat’slasttheoremhasbeenproved. For more than three and a half centuries, mathematicians — from the greatnamestothecleveramateurs—triedtoproveFermat’sfamous statement. The approach was new and involved very sophisticated theories. Finallythelong-soughtproofwasachieved. Thearithmetic theory of elliptic curves, modular forms, Galois representations, and their deformations, developed by many mathematicians, were the tools required to complete the di?cult proof. Linked with this great mathematical feat are the names of TANI- YAMA, 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 presentingtheproofofFermat’stheorem. Onthecontrary, itiswr- ten 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 N- bers and described in my 13 Lectures on Fermat’s Last Theorem (new printing, containing an Epilogue about recent results).

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### Contents

 Special Cases 3 I2 The Biquadratic Equation 11 I3 Gaussian Numbers 21 I4 The Cubic Equation 24 I5 The Eisenstein Field 41 I6 The Quintic Equation 49 17 Fermats Equation of Degree Seven 57 I8 Other Special Cases 63
 I The NonExistence of Algebraic Identities Yielding Solutions of Fermats Equation 269 J Criterion with SecondOrder Linear Recurrences 270 K Perturbation of One Exponent 272 L Divisibility Condition for Pythagorean Triples 273 Interludes 9 and 10 277 IX2 Lagrange Resolvents and Jacobi Cyclotomic Function 282 The Local and Modular Fermat Problem 287 X2 Fermat Congruence 291

 19 Appendix 71 4 Interludes 73 II2 Cyclotomic Polynomials 77 II3 Factors of Binomials 79 Algebraic Restrictions on Hypothetical Solutions 99 III2 Secondary Relations for Hypothetical Solutions 106 Germains Theorem 109 IV2 Wendts Theorem 124 Sophie Germains Primes 139 Interludes 5 and 6 143 V2 Linear Recurring Sequences of Second Order 156 Arithmetic Restrictions on Hypothetical Solutions and on the Exponent 164 VI2 Divisibility Conditions 184 VI3 Abels Conjecture 195 VI4 The First Case for Even Exponents 203 Interludes 7 and 8 213 VII2 The Cauchy Polynomials 220 Reformulations Consequences and Criteria 235 B Reformulations of Fermats Last Theorem 247 VIII2 Criteria for Fermats Last Theorem 253 B Connection with the Mobius Function 255 D Criterion with a Legendre Symbol 256 E Criterion with a Discriminant 257 F Connection with a Cubic Congruence 263 G Criterion with a Determinant 266 H Connection with a Binary Quadratic Form 267
 X3 Hurwitz Congruence 304 X4 Fermats Congruence Modulo a PrimePower 316 Epilogue 359 A The Theorem of Kummer 360 B The Theorem of Wieferich 361 C The First Case of Fermats Last Theorem for Inﬁnitely Many Prime Exponents 363 E The abc Conjecture 364 XI2 Victory or the Second Death of Fermat 366 A The Frey Curves 367 B Modular Forms and the Conjecture of ShimuraTaniyama 369 C The Work of Ribet and Wiles 373 XI3 A Guide for Further Study 375 B Expository 376 C Research 378 XI4 The Electronic Mail in Action 379 References to Wrong Proofs 381 I Papers or Books Containing Lists of Wrong Proofs 382 III Insufficient Attempts 387 General Bibliography 389 II Books Primarily on Fermat 390 III Books with References to Fermats Last Theorem 391 IV Expository Historical and Bibliographic Papers 392 V Critical Papers and Reviews 395 Name Index 396 Subject Index 404 Copyright

### About the author (2000)

Ribenboim, Queen's University, Ontario, Canada.