Fermat's theorem is proved at last . . . but what does it matter?
Sunday 27 June 1993
It had all started with Pierre de Fermat (1601-65) and a 3rd-century Greek mathematician, Diophantus of Alexandria. Besides being one of the greatest French mathematicians, Fermat was one of the great non-publishers of all time, being content to summarise his findings in scribbled comments in the margin of his copy of Arithmetics by Diophantus.
Thus, where Diophantus showed how to find triples of numbers (e g, 3, 4, 5 or 5, 12, 13) such that the squares of the first two add up to the square of the third, Fermat wrote: 'It is impossible for a cube to be written as the sum of two cubes, or a fourth power to be written as the sum of two fourth powers, or generally for any number which is a power greater than the second, to be written as a sum of two like powers.'
In symbols, he was claiming that there were no whole numbers x, y, z, and n such that x to the power n + yx to the power n = zx to the power n , for any n greater than 2. And, to his statement of the theorem, he added: 'I have a truly wonderful demonstration of this proposition which this margin is too narrow to contain.'
When Fermat made such a statement, he was eventually almost invariably proved right. The competitive style of mathematics of his time was very much to state results and challenge others to prove them, rather than spreading wisdom. By the 19th century, however, it was generally agreed that this time Fermat had got it wrong. The truth or falsity of his theorem (or conjecture, as it should properly have been called until last Wednesday) lay far deeper in the mathematical field of rational numbers than was dreamt of in the 17th century.
Fermat's 'proof', like many more recent attempts, had probably involved making a plausible, but unjustified, assumption about the structure of numbers. But for 300 years, plausible assumptions were the best anyone had to offer.
When Professor Andrew Wiles of Cambridge University and Princeton began his series of three lectures at the Newton Institute last Monday, there was a feeling that he might reveal something special. By the end of Tuesday's lecture, it was felt he was building up to something big.
'I'd heard more and more rumours,' said Professor Kenneth Ribet of Berkeley, California, 'so I brought my camera.' Had it been a video camera, it would have recorded the 40-year-old Briton ending Wednesday's lecture, having proved a major theorem in the theory of elliptic curves, then writing 'Corollary' on the blackboard, followed by Fermat's Last Theorem, and saying: 'I think I'll stop there.'
The audience, comprising many of the best mathematicians from around the world, greeted this with open-mouthed silence for about 10 seconds, followed by a burst of applause. Then, after some technical questions, there was more applause before they went for coffee.
The principal result that Professor Wiles proved was something known as Taniyama's conjecture, which for the past 10 years has been known to be inextricably linked to Fermat's Theorem. Some of the bonds between the two were discovered by Professor Ribet, whose work helped to establish that the theorem was unlikely to be an isolated quirk of numbers but, if true, fitted in with other aspects of mathematics.
'We knew that Taniyama's conjecture had to be true,' he said. 'It fitted into a whole string of interlocking conjectures. But we knew that we were unable to prove it.' Professor Wiles's proof is in fact of such complexity that it takes up 200 pages of still unpublished manuscript.
Professor John Coates of the Isaac Newton Institute described the result as the mathematical equivalent of splitting the atom, but Professor Wiles shyly said: 'Its significance is primarily symbolic'. So where does mathematics go from here, now that the Everest of Fermat has been scaled? Millions of amateur number- theorists must now find another problem. We can recommend any of the following, all still unproved:
The Goldbach Conjecture, that every even number is the sum of two prime numbers
The Prime Pairs Problem: to prove that there are infinitely many pairs of prime numbers, such as 5, 7, or 29, 31, or 101, 103 where the higher is two more than the lower
The 3N+1 problem: Start with any number; if it is even, halve it, if it is odd, multiply by three and add one; keep repeating this procedure. Prove that whatever number you started with, you will always reach the number 1
Like much of pure mathematics, neither these problems nor Fermat's Theorem has any conceivable application; but, as Professor Ribet points out, much of modern cryptography and theoretical physics uses mathematics that once had no conceivable application. 'When I went into this, I was quite proud of choosing an area that would never be applied,' he said. 'I have more perspective now . . .'
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