Over The Horizon

Fermat's Last Theorum -- At Last!

In 1637 Pierre de Fermat wrote a theorem in the margin of his copy of the works of Diophantus. He claimed to have a truly remarkable proof of the theorem that was too big to write in the margin. About 356 years later, Prof Andrew Wiles of Princeton University presented a series of three seminars at the Isaac Newton Institute of Mathematical Sciences in Cambridge. In the verbal equivalent of a marginal note, at the end of his seminars Wiles pointed out that his results prove Fermat's last theorem.

Although its proof has stumped mathematical geniuses for three centuries, Fermat's last theorem is simple in appearance. It states that for any integer n greater than 2, the equation xⁿ + yⁿ = zⁿ has no solutions where x , y, and z are positive integers. (When n equals 2, the equation is just the Pythagorean theorem concerning right-angled triangles, with many solutions.)

Fermat himself wrote down a proof for n = 4 and the n = 3 case was proved by Leonhard Euler in 1780. Proofs for certain other cases soon followed. Computer programs verified the theorem up to large numbers, and while that lent weight to the suspicion that the theorem holds in general, it didn't prove it.

There is a strong suspicion that Fermat did not have a proof of his theorem. Many mathematicians have worked at the problem, and many flawed "proofs" have been announced. Indeed, until it has been published and carefully scrutinized, it is possible that a subtle mistake will be found in Wiles's proof, although most experts in the field seem to believe that it is valid. One thing is certain: you couldn't fit Wiles's proof in the margin of a book, either -- it runs to 200 pages.

The chain of ideas culminating in Wiles's proof began with a conjecture made in 1954 by Yutaka Taniyama about mathematical objects known as "elliptic curves". He conjectured that a wide class of such curves had to have a special property: they had to be "modular". In the mid-1980s Gerhard Frey connected the Taniyama conjecture to Fermat's last theorem, and in 1987 Kenneth Ribet made the connection rigorous -- proving the Taniyama conjecture would also prove Fermat's last theorem.

Essentially, one associates an elliptic curve with each equation of Fermat's theorem. The work of Frey and Ribet showed that if an equation has a positive-integer solution, then its associated elliptic curve cannot be modular. Wiles proved the Taniyama conjecture, which forces all the curves to be modular. Therefore none of the associated equations can have positive-integer solutions, and Fermat's marginal note is vindicated.

This collection of ideas forges a link between the widely disparate fields of algebra and analysis. This aspect of Wiles's proof is more important than the knowledge that, indeed, Fermat's theorem is true.

Graham P. Collins, NZSM, New York

Graham P. Collins NZSM, New York