Fermat's Last Theorem (44 page)

Extrait d'une lettre de M. Kummer a M. Liouville, by E.E. Kummer,
J. Math. Pures et Appl.
12
(1847), 136. Reprinted in
Collected Papers
, Vol. I, edited by A. Weil, 1975, Springer.

A Number for Your Thoughts
, by Malcolm E. Lines, 1986, Adam Hilger. Facts and speculations about numbers from Euclid to the latest computers, including a slightly more detailed description of the dot conjecture.

3.1416 and All That
, by P.J. Davis and W.G. Chinn. 1985, Birkhäuser. A series of stories about mathematicians and mathematics, including a chapter about Paul Wolfskehl.

The Penguin Dictionary of Curious and Interesting Numbers
, by David Wells, 1986, Penguin.

The Penguin Dictionary of Curious and Interesting Puzzles
, by David Wells, 1992, Penguin.

Sam Loyd and his Puzzles
, by Sam Loyd (II), 1928, Barse and Co.

Mathematical Puzzles of Sam Loyd
, by Sam Loyd, edited by Martin Gardner, 1959, Dover.

Riddles in Mathematics
, by Eugene P. Northropp, 1944, Van Nostrand.

The Picturegoers
, by David Lodge, 1993, Penguin.

13 Lectures on Fermat's Last Theorem
, by Paulo Ribenboim, 1980, Springer. An account of Fermat's Last Theorem, written prior to the work of Andrew Wiles, aimed at graduate students.

Mathematics: The Science of Patterns
, by Keith Devlin, 1994, Scientific American Library. A beautifully illustrated book which conveys the concepts of mathematics through striking images.

Mathematics: The New Golden Age
, by Keith Devlin, 1990, Penguin. A popular and detailed overview of modern mathematics, including a discussion on the axioms of mathematics.

The Concepts of Modem Mathematics
, by Ian Stewart, 1995, Penguin.

Principia Mathematica
, by Betrand Russell and Alfred North Whitehead, 3 vols, 1910, 1912, 1913, Cambridge University Press.

Kurt Gödel, by G. Kreisel, Biographical Memoirs of the Fellows of the Royal Society, 1980.

A Mathematician's Apology
, by G.H. Hardy, 1940, Cambridge University Press. One of the great figures of twentieth-century mathematics gives a personal account of what motivates him and other mathematicians.

Alan Turing: The Enigma of Intelligence
, by Andrew Hodges, 1983, Unwin Paperbacks. An account of the life of Alan Turing, including his contribution to breaking the Enigma code.

Yutaka Taniyama and his time, by Goro Shimura,
Bulletin of the London Mathematical Society
21
(1989), 186–196. A very personal account of the life and work of Yutaka Taniyama.

Links between stable elliptic curves and certain diophantine equations, by Gerhard Frey,
Ann. Univ. Sarav. Math. Ser.
1
(1986), 1–40. The crucial paper which suggested a link between the Taniyama–Shimura conjecture and Fermat's Last Theorem.

Genius and Biographers: the Fictionalization of Evariste Galois, by T. Rothman,
Amer. Math. Monthly
89
(1982), 84–106. Contains a detailed list of the historical sources behind Galois's biographies, and discusses the validity of the various interpretations.

La vie d'Evariste Galois, by Paul Depuy,
Annales Scientifiques de l'Ecole Normale Supérieure
13
(1896), 197–266.

Mes Memoirs
, by Alexandre Dumas, 1967, Editions Gallimard.

Notes on Fermat's Last Theorem
, by Alf van der Poorten, 1996, Wiley. A technical description of Wiles's proof aimed at mathematics undergraduates and above.

An elementary introduction to the Langlands programme, by Stephen Gelbart,
Bulletin of the American Mathematical Society
10
(1984), 177–219. A technical explanation of the Langlands programme aimed at mathematical researchers.

Modular elliptic curves and Fermat's Last Theorem, by Andrew Wiles,
Annals of Mathematics
141
(1995), 443–551. This paper includes the bulk of Wiles's proof of the Taniyama–Shimura conjecture and Fermat's Last Theorem.

Ring-theoretic properties of certain Hecke algebras, by Richard Taylor and Andrew Wiles,
Annals of Mathematics
141
(1995), 553–572. This paper describes the mathematics which was used to overcome the flaws in Wiles's 1993 proof.

You can find a set of websites about Fermat's Last Theorem on Simon Singh's website:

[http://www.simonsingh.com]

The pagination of this electronic edition does not match the edition from which it was created. To locate a specific passage, please use the search feature of your e-book reader.

Page numbers in
italic
refer to illustrations

Abel, Niels Henrik 3

absolute proof 21–7, 147

absurdities, mathematical 143, 341

Academy of Sciences, French 119, 238

prize for proving Fermat's Last Theorem 120–28

ACE (Automatic Computing Engine) 175

Adleman, Leonard 104

Adler, Alfred 2

Agnesi, Maria 109–10, 111, 119

Alexandria 47–9, 57–8, 109

Alexandrian Library 48–9, 57–8

Algarotti, Francesco 112

algorithms 81

amicable numbers 62–3

Anglin, W. S. 77

Annals of Mathematics
303

April fool e-mail 293–5

Arago, François 79

Arakelov, Professor S. 254

Archimedes 48, 112

Aristotle 59

arithmetic algebraic geometrists 254–5

Arithmetica
(Diophantus) 42, 57, 58, 60, 61, 62

Clément-Samuel Fermat's edition
68–9
, 70

and elliptic equations 184

Fermat's marginal notes 62, 66–7, 70, 89

Latin translation
56
, 61, 62

and Pythagorean triples 65

axioms 21, 149, 155, 156

of arithmetic 342–3

consistency of 159–60

Babylonians 7–8, 20, 59

Bachet de Méziriac, Claude Gaspar 61–2

Latin translation
of Arithmetica 56
, 61, 62

Problèmes plaisants et delectables
61

weighing problem 61, 337–8

Barnum, P. T. 138

Bell, Eric Temple 6, 30, 33, 39, 73, 115

Bernoulli family 79–80

birthdays, shared, probability of 44–5

Bombelli, Rafaello 93–4

Bonaparte, Napoleon 117, 124, 232, 234

Bourg-la-Reine 232, 234, 238

Brahmagupta 59

bridges, mathematical 212

Bulletin of the London Mathematical Society
207

calculus 18, 46–7

Cantor, Georg 101–2

Cardano, Girolamo 40–41

Carroll, Lewis 138

Cauchy, Augustin Louis 120–28,
122
, 238, 239

chessboard, mutilated, problem of 24–6

Chevalier, Auguste 245, 248

Chudnovsky brothers 51

Churchill, Sir Winston Leonard Spencer 174

cicadas, life-cycles 106–7

Circle Limit IV
(Escher)
200
, 201

City of God, The
(St Augustine) 12

Clarke, Arthur C. 23

clock arithmetic 185–8

closed groups 250–51

Coates, John 180,
182
, 183, 189, 211,226,229, 260, 266,270, 284, 303–4

code breaking 103–5, 168, 170–75

Cohen, Paul 162–3

Colussus (computer) 175

commutative law of addition 149

completeness 91–2, 149–50, 160

complex numbers 95, 126

computers

early 175, 176

unable to prove Fermat's Last Theorem 177–8

unable to prove Taniyama–Shimura conjecture 231

conjectures 72

unifying 305

Constantinople 60

continuum hypothesis 163

contradiction, proof by 49–50, 53–4, 155

Conway, Professor John H. 291

Coolidge, Julian 39

cossists 40

counting numbers 11

Cretan paradox 161

Croton, Italy 9, 27–8

cryptography 103–5, 168, 170–75

crystallography 199, 310

cubic equations 237

Curiosa Mathematica
(Dodgson) 138

Cylon 27–8

d'Alembert, Jean Le Rond 96

Dalton, John 22

Darmon, Henri 294, 295

Deals with the Devil
74

defective numbers 11

slightly 13

Descartes, René 41, 42, 63, 249

Deuring 192

Devil and Simon Flagg, The
37, 74

d'Herbinville, Pescheux 243, 247, 248

Diderot, Denis 82–3

differential geometry 254, 256

Diffie, Whitfield 104

Digby, Sir Kenelm 38, 64

Diophantine problems 57

Diophantus of Alexandria 55, 57

riddle of his age 55, 57, 336–7

Diophantus' Arithmetica Containing Observations by P. de Fermat 68–9
, 70

Dirichlet, Johann Peter Gustav Lejeune 116, 127, 188

disorder parameters 140–42

Disquisitiones arithmeticae
(Gauss) 115

Dodgson, Reverend Charles 138

domino effect 232

dot conjecture problem 128–9, 339–40

du Motel, Stéphanie-Félicie Poterine 243, 248

Dudeney, Henry 138

Dumas, Alexandre 241–2

E
-series 188–9, 204–5, 211, 251–3

École Normale Supérieure 240

École Polytechnique 113–14, 236

economics, and calculus 46

Eddington, Sir Arthur 133

Egyptians, ancient 7–8

Eichler 195

Eiffel Tower 119

Einstein, Albert 17, 18, 110

electricity, and magnetism 204–5

Elements
(Euclid) 49, 53, 55, 125

elephant and tortoise fable 160

Elkies, Noam 179, 293–5

elliptic curves 183

elliptic equations 183–5, 187–9, 202

families of 261, 265

Frey's elliptic equation 216–19, 221–2

and modular forms 202, 204–5, 209–15, 305

Enigma code 168–74

Epimenides 161

Escher, Mauritz 201

Euclid

infinite number of Pythagorian triples proof 65, 338

infinity of primes proof 100–101

and perfect numbers 13

proves that 2 is irrational 53. 334–6

and
reductio ad absurdum
49, 53–4

unique factorisation proof 125

Euler, Leonhard 33, 63,
76

attempts to solve Fermat's Last Theorem 88–9, 90, 96

blindness and death 96–8

forsakes theology 79–80

and Königsberg bridge puzzle 83–5

phases of the moon algorithm 81–2, 97

proves existence of God 82–3

proves network formula 85–8

solves prime number theorem 70–71

Euler's conjecture 178–9

Evens, Leonard 284

Eves, Howard W. 225

excessive numbers 11

slightly 13–14

factorisation, unique 125–6

Faltings, Gerd 255–6, 257, 300

Fermat, Clément-Samuel 67, 70

Fermat, Pierre de
36

amateur mathematician 39

Arithmetica
61, 62, 65–7

calculus 46–7

career in civil service 37–9, 60–61

death 67

education 37

and elliptic equations 184

and Father Mersenne 41–2

ill with plague 38–9

observations and theorems 70–73

probability theory 43–4, 45–6

reluctant to reveal proofs 42

Fermat's Last Theorem

challenge of 72–4

computers unable to prove 177–8

Miyaoka's ‘proof 254–7

partial proofs by computer 177

Germain's method 115–17

n
= 3 (Euler) 90, 96, 99

n
= 4 (Fermat) 89–90, 98–9

n
= 5 (Dirichlet and Legendre) 116

n
= 7 (Lamé) 116

n
= irregular prime (Kummer and Mirimanoff) 176–7

publication of 70

and Pythagoras' equation 32, 65–7

scepticism as to existence of proof 128

simplicity of statement 6, 73

and Taniyama–Shimura conjecture 216–19, 221–3, 266

and undecidability 163–4, 166

why called ‘Last' 72

Wiles's proof
see
Wiles, Andrew

Fermatian triple 66

finite simple groups Flach, Matheus 260

four-colour problem 319–26

four-dimensional shapes 255–6

four-dimensional space 201

Fourier, Jean Baptiste Joseph 239

‘14–15' puzzle 139–42, 219

fractions 11, 53, 90–91

Frege, Friedrich Ludwig Gottlob 150, 152, 154

Frey, Gerhard 215–19

Frey's elliptic equation 216–19, 221–2

friendly numbers 62–3

fundamental particles of matter 22–3

fundamental theorem of arithmetic 125

fundamental truths 148–9

Furtwängler, Professor P. 157, 159

Galileo Galilei 39

Galois, Évariste 3,
233

birth 232

duel with d'Herbinville 243, 247, 248

education 234–6, 240

final notes 243,
244
, 245,
246
, 247, 248

funeral 247–8

and group theory 250–51, 252–3

and quintic equations 238, 239–40, 245, 248–9

revolutionary career 238–9, 240–43

game theory 167–8, 343–4

Gardner, Martin 63, 146

Gauss, Carl Friedrich 114–15, 116, 117–18, 119, 179

geometry 7–8, 322

rubber-sheet 322

Gerbert of Aurillac 60

Germain, Sophie 107,
108
, 111–14, 119

career as a physicist 118–19

and Évariste Galois 240–41