# 13 Lectures on Fermat's Last Theorem

Springer Science & Business Media, Dec 18, 1979 - Computers - 302 pages
Fermat'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 Index of Names 299 Subject Index 305 Copyright