Fermat's Last Theorem (45 page)

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

points (dice game) 43

polynomials 237

Portraits from Memory
(Russell) 160

prime numbers 70–71

almost primes 308

and Fermat's Last Theorem 99–100

Germain primes 116

infinity of 100–101, 102–3

irregular primes 126–7, 177

practical applications 103–7

333,333,331 not prime 178

twin primes 308

Principia Mathematica
(Russell and Whitehead) 156–7

probability 43–7

counter-intuitive 44–5

Problèmes plaisants et delectables
(Bachet) 61

puzzles, compendiums of 138

Pythagoras

abhors irrational numbers 50, 54–5

at Croton 9–10, 27–8

death 28

and mathematical proof 26

and musical harmony 14–17

and perfect numbers 12–13

and study of numbers 7

travels 7–8

Pythagoras' equation 28

‘cubed' version 30–32

and Fermat's Last Theorem 32, 65–6

whole number solutions 28–30

Pythagoras' theorem 6–7, 19–20, 26, 333–4

Pythagorean Brotherhood 9–11, 13, 27–8, 49, 50, 108

Pythagorean triples 28–30, 65, 338

quadratic equations 236–7

quantum physics 162

quartic equations 237

quintic equations 237–8, 239–40, 245, 248–9

Ramanujan, Srinivasa 3

Raspail, François 242–3

rational numbers 11

rearrangement of equations 216

recipes, mathematical 8, 237

reductio ad absurdum
49–50, 53–4

reflectional symmetry 196

Reidemeister, Kurt 142

religion, and probability 46

Reynolds 323

Ribenboim, Paulo 144

Ribet, Ken
220
, 229, 267, 270–71, 272,
276
, 288–9, 304

Fermat Information Service 282

and significance of Taniyama–Shimura conjecture 221–3

Riemann hypothesis 73

river ratio 17–18

Rivest, Ronald 104

Rosetta stone 212

Rossi, Hugo 46–7

rotational symmetry 195–6

Rubin, Professor Karl 268–9, 300

Russell, Bertrand, 22, 44, 147,
153
, 160

Russell's paradox 152, 154–7

St Augustine (of Hippo) 12

Sam Loyd and his Puzzles: An Autobiographical Review
138

Samos, Greece 8–9

Sarnak, Peter 285–6, 291

Schlichting, Dr F. 144–6

scientific proof 21–2

scientific theories 22–3

scrambling and unscrambling messages 103–5, 168, 170–75

Segre 314

Selmer groups 287

Shamir, Adi 104

Shimura, Goro
193
, 191–5, 202, 203,
206

relationship with Taniyama 205, 207, 209

and Taniyama–Shimura conjecture 209–10, 272, 274

Shimura-Taniyama conjecture
see

Taniyama–Shimura conjecture Silverman, Bob 284

Sir Isaac Newton Institute, Cambridge 4–5, 266

Sir Isaac Newton's Philosophy Explain'd for the Use of Ladies
(Algarotti) 112

6, perfection of 11–12

Skewes, S. 179–80

Skewes's number 180

sociable numbers 63–4

Socrates 109

Somerville, Mary 113

square, symmetries of 195–6

square-cube sandwiches 64, 184

square root of one 93

square root of two 53–4, 91–2, 312–4

strings

and particles 23

vibrating 15–17,
16

Suzuki, Misako 207, 208

symmetry 195–202

Taniyama, Yutaka
190
, 191–5, 202, 203

death 205, 207–8

influence of 209

and Taniyama–Shimura conjecture 202, 204–5

Taniyama–Shimura conjecture 205, 209–15

and Fermat's Last Theorem 216–19, 221–3

Wiles and 215, 223, 225–31, 232, 258–61, 263–5, 274, 304

Taniyama-Weil conjecture
see

Taniyama–Shimura conjecture

Tartaglia, Niccolò 40–41

Taylor, Richard 285,
292
, 293, 296, 297, 299–300

Thales 26

Theano 9–10, 107–8

theorems 21, 71–2

Theory of Games and Economic Behaviour, The
(von Neumann) 167

13 Lectures on Fermat's Last Theorem
(Ribenboim) 144

Thomson, J. J. 22

three-body problem 81

threeness 152

tiled surfaces, symmetry of 196–9

Titchmarsh, E. C. 166

Tokyo, international symposium (1955) 203

translational symmetry 196–7

trichotomy, law of 148

truels 167, 343

Turing, Alan Mathison 167–176

uncertainty principle 161–2

undecidability theorems 159–63

von Neumann, John 159, 167

Wagstaff, Samuel S. 176

Wallis, John 38, 42, 64

weighing problem 61, 337–8

Weil, André 160, 210

Weil conjecture
see
Taniyama–Shimura conjecture Weyl, Hermann 149

Whitehead, Alfred North 156

whole numbers 11

Wiener Kreis
(Viennese Circle) 157

Wiles, Andrew
xviii, 181, 224, 276, 302

adolescence and Fermat's Last Theorem 5–6, 33, 77–8

graduate student days 180–81, 183

tackles elliptic equations 183, 184–5, 188, 189

and Taniyama–Shimura conjecture 215, 223, 225–31, 232, 258–61, 263–5, 274

uses Galois's groups 251–3, 258, 296

announces proof of Fermat's Last

Theorem 1–2, 5, 33–5,
34
, 266–72

reaction of media 272–4

mathematical celebrity 274, 290–91

submits proof for verification 277–9

proof flawed 279–91, 293, 296

proof revised 296–300

proof published 304–5

wins Wolf Prize 308

collects Wolfskehl Prize 308

and the future 309

Wiles, Nada 230, 265, 281, 298–9

Wolf Prize 306

Wolfskehl, Paul
132
, 133–5

Wolfskehl Prize 135–7, 143–6, 268

Zagier, Don 254

zero, function of 58–9

FERMAT'S LAST THEOREM

Simon Singh received his PhD from the University of Cambridge. A former BBC producer, he directed the BAFTA award-winning documentary film
Fermat's Last Theorem
and wrote the best selling book of the same name. He is also the author of
The Code Book
and
Big Bang.

The Code Book
Big Bang

Fourth Estate

An imprint of HarperCollins
Publishers

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Hammersmith

London W6 8JB

www.harpercollins.co.uk

First published in paperback by Fourth Estate in 2002 (reprinted 4 times)

First published in Great Britain in 1997 by Fourth Estate

Copyright © 1997 by Simon Singh

Foreword copyright © 1997 by John Lynch

Line illustrations by Jed Mugford

The right of Simon Singh to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

A catalogue record for this book is available from the British Library.

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