13 Lectures on Fermat's Last Theorem

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Springer Science & Business Media, Dec 18, 1979 - Computers - 302 pages
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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
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Ribenboim, Queen's University, Ontario, Canada.

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