## Quick Reference

(1953–) British mathematician

The son of a theology professor, Wiles was educated at Cambridge University, where he gained his PhD in 1980. He immediately took up an appointment as professor of mathematics at Princeton.

Wiles worked on the most famous of all mathematical problems, namely, ‘Fermat's last theorem’ (FLT). Pierre Fermat claimed in 1637 that he had proved that there are no numbers *n* ≥ 2 such that:

*a ^{n}* +

*b*=

^{n}*c*

^{n}It was, of course, well known that where *n* = 2 there were many solutions to the equation as when *a* = 3, *b* = 4, and *c* = 5. Fermat had merely stated the theorem, as was his custom, in the margins of a book, in this case the Arithmetica of Diophantus. He simply added that there was insufficient room in the margin to record the details of the proof.

As over the years all Fermat's other marginalia have turned out to be accurate and proofs found, mathematicians were optimistic that the one unproved proposition – the last theorem – would also succumb. Yet 300 years later not only had no proof been found, but mathematicians seemed unaware in which direction a proof could be found. They could, of course, simply show the proposition to be false by finding an *n* ≥ 3 which does satisfy the equation. But that was going to be no easy matter either, for it had been shown that any such number *n* must be very large – by 1992 all exponents up to 4 million had been tested and failed.

An alternative approach was suggested by some work in 1954 on elliptic curves by the Japanese mathematician Yutaka Taniyama. An elliptic curve is a set of solutions to an equation relating a quadratic in one variable to a cubic in another as in:

*y*^{2} = *ax*^{3} + *bx*^{2} + *cx* + *d*

The Taniyama conjecture asserts that associated with every elliptical curve was a function with certain very precise specific properties. The German mathematician Gerhard Frey argued in 1985 that the Taniyama conjecture had important implications for Fermat's last theorem. He demonstrated that any possible solution to the theorem would give rise to a class of elliptical curves, referred to as *Frey curves*, which could not satisfy the conditions of the Taniyama conjecture. Thus a proof of the Taniyama conjecture would show that there could be no solutions to Fermat's last theorem.

In 1986 Wiles set out to show that Frey curves could not exist. Seven years later he had established a 200-page proof, which he revealed to the public for the first time at a mathematical conference in Cambridge, England, in 1993. Although Wiles's proof made headline news around the world it soon became evident that gaps still existed. After a further year the gaps had been eliminated and the 200-page paper had been accepted for publication.

*Subjects:*
Mathematics.