One of the most impressive feats for a mathematician is to answer one of the
subject’s big riddles – problems that have baffled the best intellects for
centuries. Most problems of this kind are named after whoever first posed
them – examples are the Riemann hypothesis, the Poincare conjecture, the
Kepler problem. One of the most notorious is Fermat’s last theorem, which
has defied the best mathematicians for more than 350 years.
But last week, Andrew Wiles of Princeton University announced a 1000-page
proof at a conference in Britain. The announcement came at the end of a
lecture entitled ‘Modular forms, elliptic curves, and Galois
representations’. But the packed audience at the Isaac Newton Institute for
Mathematical Sciences in Cambridge strongly suspected he had Fermat up his
sleeve. They were right.
Pierre de Fermat (1601-1665) was a French lawyer whose hobby was
mathematics. A century before Newton, he worked out many of the basic ideas
of calculus. But his most significant achievements were in number theory, a
branch of mathematics that deals with properties of whole numbers. Number
theory is one of the most difficult branches of mathematics, a field in
which conjectures – unproved guesses and number patterns – are ten a penny,
but proofs are often elusive.
Fermat’s last theorem – so-called because it was for many years the sole
assertion of his that had neither been proved nor disproved – is just such a
guess. The ancient Greeks knew that there are an infinite number of
Pythagorean triples – whole numbers that can form the sides of a right
triangle. Pythagoras’s theorem tells us that such numbers x, y, z must
satisfy the equation x2 + y2 = z2
Well-known examples are 32 + 42 = 52
and 52 + 122 = 132
. Fermat wondered if the same kind of relationship could occur with
cubes, fourth powers, and so on. He convinced himself that it could not. In
the margin of his copy of the Arithmetica of Diophantos, an early Greek
treatise on equations, Fermat wrote: ‘To resolve a cube into the sum of two
cubes, a fourth power into two fourth powers, or in general any power higher
than the second into two of the same kind, is impossible; of which fact I
have found a remarkable proof. The margin is too small to contain it.’
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Fermat was asserting that the equation xn + yn = zn
has no integer solutions when n => 3, other than
trivial solution in which either x, y or z is zero. For a long time, this
conjecture seemed no more than a historical curiosity – it has no direct
applications outside pure mathematics – but it was such a simple question
that mathematicians found themselves irresistibly drawn to it.
Fermat’s ‘remarkable proof’ has never been found, and experts generally
believe that whatever Fermat had in mind must have contained an error. Few
now think that the techniques available in Fermat’s time can possibly lead
to a proof.
More than 200 years elapsed before the first really big inroad on the
theorem was made, by Ernst Kummer. He developed an idea of Fermat’s,
introducing ‘algebraic numbers’, which are more general than ordinary
integers. By devising an entirely new theory of ‘ideals’, he was able to
prove that Fermat was right for any power n =
