Fermat's Last Theorem
Page 27
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.
Chapter 6
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.
Chapter 7
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]
Index
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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
relationship with Gauss 117–18, 119
strategy for Fermat’s Last Theorem 115–17
Gibbon, Edward 109
Globe, Le 239
Gödel, Kurt 146, 157, 158, 159
undecidable statements 159–63
Goldbach, Christian 90
Gombaud, Antoine 43–4
Government Code and Cypher School 170–75
gravity, theories of 18, 23
group theory 250–51
Grundgesetze der Arithmetik (Frege) 152, 154
Guardian 272
hammers, harmony of 15
Hardy, G.H. 1, 2–4, 49–50, 165, 166, 179–80, 191
Riemann hypothesis telegrams 73
Hecke algebras 299–300
Hein, Piet 277
Heisenberg, Werner 162
Hellman, Martin 104
Hermite, Charles 3
hieroglyphics 212
Hilbert, David 101–3, 147, 151, 157
and basic axioms 149–50
and Fermat’s Last Theorem 226–7, 268
23 problems 150, 160, 162, 163
Hilbert’s Hotel 102–3
Hippasus 54
History of Mathematics (Montucla) 112
Hodges, Andrew 176
Hypatia 109, 111
hyperbolic space 201
Iamblichus 14–15
Illusie, Luc 278, 281
imaginary numbers 90, 93–6, 125–6
induction, proof by 231–2, 322–3
infinite descent, method of 90–91
infinity 59, 101–3, 177–8
International Congress of Mathematicians Berkeley (1986) 221, 222
Paris (1900) 150
intuition, and probability 44–5
invariants 141, 142, 219
Inventiones Mathematicae 277
irrational numbers 50, 54, 90–92
Iwasawa theory 259, 260, 296, 297–8
Journal de Mathématique pures et appliquées 248
Kanada, Yasumasa 51
Katz, Nick 262, 263–5, 278–80, 281
knot invariants 142, 219
Kolyvagin–Flach method 259–61, 263–5, 279–80, 281, 293, 297–8
Königliche Gesellschaft der Wissenschaften 135–7, 277
Königsberg bridge puzzle 83–5
Kovalevsky, Sonya 111
Kronecker, Leopold 50
Kummer, Ernst Eduard 123–8, 124, 134–5, 176–7
L-series 188
Lagrange, Joseph-Louis 96, 114, 239
Lamé, Gabriel 116, 120–27, 121
Landau, Edmund 110, 143–4
Langlands, Robert 213, 306
Langlands programme 213–14, 254
Last Problem, The (Bell) 6, 30, 33, 73
Le Blanc, Antoine-August 114
see also Germain, Sophie Legendre, Adrien-Marie 116
Leibniz, Gottfried 93
liar’s paradox 161
Libri-Carrucci dalla Sommaja, Count Guglielmo 113, 241
light, nature of 204–5
limping triangles 65
Liouville, Joseph 124–5, 248, 249
Lipman, Joseph 283
Littlewood, John Edensor 179
Lodge, David 177–8
logic, mathematical 148–9
logicians 148–9, 162
loopiness, in rivers 17–18
Loyd Sam 138–42
Loyd’s puzzle see ‘14–15’ puzzle
lyre, tuning strings on 14–17
M-series 201–2, 204–5, 211, 251–3
magnetism, and electricity 204–5
Mahler 314
Mathematical Magic Show (Gardner) 63
mathematical proof 20–21, 23–6
Mathematician’s Apology, A (Hardy) 2–3, 49–50, 166
mathematicians
collaboration amongst 4–5
and compulsion of curiosity 164–6
in India and Arabia 58–60, 93
mathematical life 2–4
require absolute proof 147–8
secretive nature 40–41
self-doubt of 78–9
youthfulness 3
mathematics
contradictory nature of 152, 154–7
foundation for science 26–7
objective subject 28
relationship with science 17, 18
in seventeenth century 39–40
Mathematics of Great Amateurs (Coolidge) 39
Mathematische Annalen 192
Mazur, Barry 211–12, 221, 265, 267, 270, 271, 277
Mersenne, Marin, Father 40–42
Method, The (Heiberg) 48
meticulous librarian, tale of 154–5
Milo 9, 27–8
Mirimanoff, Dimitri 177
Miyaoka, Yoichi 254, 256–7
Miyaoka inequality 256
modular forms 195, 199–202
and elliptic equations 202, 204–5, 209–15
Monde, Le 272
Montucla, Jean-Étienne 112
moon, predicting phases of 81–2
Moore, Professor L. T. 47
Mozans, H.J. 119
musical harmony, principles of 14–17
My Philosophical Development (Russell) 154
natural numbers 91
negative numbers 90–94
network formula 85–8
New York, subway graffiti 257
New York Times 254, 272–3, 282
Newton, Isaac 18, 47, 80, 81
Nixon, Richard Milhous 46–7
Noether, Emmy 110–11
nothingness, concept of 59
number line 92, 94–5, 185–6
numbers
definition of 150, 152
relationships between 11
numerals, Indo-Arab 59–60
Oberwolfach symposium (1984) 215–19, 221
Olbers, Heinrich 115
order and chaos 17
overestimated prime conjecture 179
Paganini, Nicolò 63
parallelism, philosophy of 254, 257
parasites, life-cycles 106–7
particle physics 22–3
Pascal, Blaise 40, 43–4, 45–6
Penrose, Roger 198
Penrose tilings 198–9
People 274, 290–91
perfect numbers 11–13
philosopher, word coined by Pythagoras 10
pi (π) 17–18, 50–53, 166
Picturegoers, The (Lodge) 177–8
Pillow Problems (Dodgson) 138
Pinch, Richard 285
Plato 109
Poges, Arthur 37, 74
Poincaré, Jules Henri 199