## 13 Lectures on Fermat's Last TheoremFermat's problem, also ealled Fermat's last theorem, has attraeted the attention of mathematieians far more than three eenturies. Many clever methods have been devised to attaek the problem, and many beautiful theories have been ereated with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topie is presented in the form of leetures, where I survey the main lines of work on the problem. In the first two leetures, there is a very brief deseription of the early history , as well as a seleetion of a few of the more representative reeent results. In the leetures whieh follow, I examine in sue eession the main theories eonneeted with the problem. The last two lee tu res are about analogues to Fermat's theorem. Some of these leetures were aetually given, in a shorter version, at the Institut Henri Poineare, in Paris, as well as at Queen's University, in 1977. I endeavoured to produee a text, readable by mathematieians in general, and not only by speeialists in number theory. However, due to a limitation in size, I am aware that eertain points will appear sketehy. Another book on Fermat's theorem, now in preparation, will eontain a eonsiderable amount of the teehnieal developments omitted here. It will serve those who wish to learn these matters in depth and, I hope, it will clarify and eomplement the present volume. |

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### Contents

The Early History of Fermats Last Theorem | 1 |

2 Early AttemptsEC | 4 |

3 Rummers Monumental Theorem | 6 |

4 Regular Primes | 9 |

5 Kummers Work on Irregular Prime Exponents | 10 |

6 Other Relevant Results | 12 |

7 The Golden Medal and the Wolfskehl Prize | 13 |

Bibliography | 16 |

4 Fermats Theorem and the Mersenne Primes | 154 |

5 Summation Criteria | 155 |

6 Fermat Quotient Criteria | 159 |

Bibliography | 161 |

The Power of Class Field Theory | 165 |

2 Kummer Extensions | 167 |

3 The Main Theorems of Furtwangler | 168 |

4 The Method of Singular Integers | 170 |

Recent Results | 19 |

2 Explanations | 21 |

Bibliography | 29 |

BK Before Kummer | 35 |

1 The Pythagorean Equation | 36 |

2 The Biquadratic Equation | 37 |

3 The Cubic Equation | 39 |

4 The Quintic Equation | 45 |

5 Fermats Equation of Degree Seven | 46 |

Bibliography | 47 |

The Naive Approach | 51 |

2 Sophie Germain | 54 |

3 Congruences | 57 |

4 Wendts Theorem | 61 |

5 Abels Conjecture | 63 |

6 Fermats Equation with Even Exponent | 65 |

7 Odds and Ends | 69 |

Bibliography | 70 |

Kummers Monument | 75 |

2 Basic Facts about the Arithmetic of Cyclotomic Fields | 77 |

3 Kummers Main Theorem | 82 |

Bibliography | 90 |

Regular Primes | 93 |

2 Bernoulli Numbers and Kummers Regularity Criterion | 99 |

3 Various Arithmetic Properties of Bernoulli Numbers | 103 |

4 The Abundance of Irregular Primes | 106 |

5 Computation of Irregular Primes | 107 |

Bibliography | 111 |

Kummer Exits | 115 |

2 The Jacobi Cyclotomic Function | 117 |

3 On the Generation of the Class Group of the Cyclotomic Field | 119 |

4 Kummers Congruences | 120 |

5 Kummers Theorem for a Class of Irregular Primes | 126 |

6 Computations of the Class Number | 130 |

Bibliography | 134 |

After Kummer a New Light | 139 |

2 The Theorem of Krasner | 148 |

3 The Theorems of Wieferich and Mirimanoff | 151 |

5 Hasse | 172 |

6 The pRank of the Class Group of the Cyclotomic Field | 178 |

7 Criteria for pDivisibility of the Class Number | 184 |

8 Properly and Improperly Irregular Cyclotomic Fields | 188 |

Bibliography | 193 |

Fresh Efforts | 199 |

2 Euler Numbers and Fermats Theorem | 202 |

3 The First Case Is True for Infinitely Many Pairwise Relatively Prime Exponents | 205 |

4 Connections between Elliptic Curves and Fermats Theorem | 207 |

5 Iwasawas Theory | 209 |

6 The Fermat Function Field | 211 |

7 Mordells Conjecture | 214 |

8 The Logicians | 216 |

Bibliography | 218 |

Estimates | 225 |

2 Estimates Based on the Criteria Involving Fermat Quotients | 229 |

3 Thue Roth Siegel and Baker | 232 |

4 Applications of the New Methods | 237 |

Bibliography | 241 |

Fermats Congruence | 245 |

2 The Local Fermats Theorem | 251 |

3 The Problem Modulo a PrimePower | 253 |

Bibliography | 260 |

Variations and Fugue on a Theme | 263 |

2 Variation II In the Tone of Entire Functions | 265 |

3 Variation III In the Theta Tone | 266 |

4 Variation IV In the Tone of Differential Equations | 270 |

5 Variation V Giocoso | 271 |

6 Variation VI In the Negative Tone | 272 |

7 Variation VII In the Ordinal Tone | 273 |

9 Variation IX In the Matrix Tone | 275 |

10 Fugue In the Quadratic Tone | 277 |

Bibliography | 286 |

Epilogue | 291 |

299 | |

305 | |

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### Common terms and phrases

Akad algebraic integers algebraic number field Amer angew arithmetic Berlin Bernoulli numbers C. R. Acad Chelsea Publ class field theory class group class number coefficients complex numbers computations conjecture criterion cyclotomic field cyclotomic integers defined denote dernier theoreme diophantine equations distinct prime divisors elliptic curve Euler exist infinitely Fermat quotient Fermat's equation Fermat's last theorem Fermat's theorem Fermatsche Vermutung Fermatschen finitely formula fractional ideals function Furtwangler Hence ideal class Inkeri irregular primes Kummer's congruences Lecture Lehmer Lemma Math Mersenne methods Mirimanoff mod q modp modp2 Morishima multiple natural numbers nombres nontrivial solution nonzero integers Number Theory odd prime p-adic pairwise relatively prime Paris polynomials prime exponents prime factors prime ideal primitive root modulo Proc proof of Fermat's pth power real number reine relatively prime relatively prime integers result satisfied theoreme de Fermat trivial solution Uber units Vandiver Wiss