Fermat’s Last Theorem for Amateurs

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Springer Science & Business Media, Feb 11, 1999 - Mathematics - 407 pages
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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

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About the author (1999)

Ribenboim, Queen's University, Ontario, Canada.

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