## Fermat’s Last Theorem for AmateursItisnowwellknownthatFermat’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 |

387 | |

General Bibliography | 389 |

390 | |

391 | |

392 | |

395 | |

396 | |

404 | |

### Common terms and phrases

algebraic Amer assume C. R. Acad Chapter coefficients congruence contradiction deﬁned dernier th´eor`eme diophantine equations distinct prime factors elliptic curves equivalent exist integers exist nonzero integers exponent Fermat’s last theorem Fermat’s theorem Fermatschen ﬁrst G Zp[X Galois group Gaussian integers Gaussian periods gcd(a hence hypothesis inﬁnite integers x,y,z Jacobi symbol l’´equation L’Interm Legendre Lemma Math mod p2 mod pm+1 mod q modp modular forms modulo q multiple nontrivial solution nonzero relatively prime nth roots number of solutions odd prime p-adic pairwise relatively prime Paris Paul Tannery polynomial Pomey positive integers prime factors prime number proof of Fermat’s proved Pures Appl q divides quadratic residue rational numbers Reine Angew relatively prime integers reprinted result solutions in integers th´eor`eme de Fermat theorem is true Uber Vandiver x,y,z are nonzero