# A curvy path leads to Fermat's last theorem.

After more than 300 years, Fermat's last theorem may finally live up to its common designation as a theorem. In a dramatic announcement that caught the mathematical community completely by surprise, Andrew Wiles of Princeton University revealed last week that he had proved major parts of a significant conjecture in number theory, These results, in turn, establish the truth of Fermat's famous, devilishly simple conjecture.

"It's an amazing piece of work," says Peter C. Sarnak, one of Wiles' Princeton colleagues. "The proof hasn't been totally checked, but it's very convincing?"

Pierre de Fermat's last theorem goes back to the 17th century, when the French jurist and mathematician asserted that for any whole number n greater than 2, the equation [x.sup.n]+[y.sup.n] = [z.sup.n] has no solution for which x, y, and z are all whole numbers greater than zero.

Fermat scribbled his conjecture in the margin of a page in a mathematics book he was reading. Then, in a tantalizing sentence that was to haunt mathematicians for centuries to come, he added that although he had a wonderful proof of the theorem, he didn't have room to write it.

After Fermat died, scholars could find no trace of the proof in any of his papers. Later, mathematicians proved the conjecture for the exponent n = 3 and solved several other special cases. Last year, a massive computer-aided effort by J.P. Buhler of Reed College in Portland, Ore., and Richard E. Crandall of NeXT Computer Inc., in Redwood City, Calif., verified Fermat's last theorem for exponents up to 4 million.

Meanwhile, mathematicians had picked up some valuable hints of a potential avenue to a general proof that the conjecture is true. In the mid-1980s, Gerhard Frey of the University of the Saarlands in Saarbrucken, Germany, unexpectedly uncovered an intriguing link between Fermat% conjecture and a seemingly unrelated branch of mathematics. He found a way to express Fermat's last theorem as a conjecture about elliptic curves - equations generally written in the form [y.sup.2] = [X.sup.3] + [ax.sup.2] + bx + c, where a, b, and c are constants.

This brought Fermat's problem into an area of mathematics for which mathematicians had already developed a wide range of techniques for solving problems. A number of mathematicians, including Barry Mazur of Harvard University and Kenneth A. Ribet of the University of California, Berkeley, followed up Frey's surprising insight with additional results that ultimately tied Fermat's last theorem to a central conjecture in number theory (SN: 6/20/87, p.397).

Named for Japanese mathematician Yutaka Taniyama, this conjecture concerns certain characteristics of elliptic curves. A proof of this conjecture would automatically imply that Fermat's last theorem must be true.

Starting about five years ago, Wiles took on the extremely challenging, highly technical task of proving the Taniyama conjecture itself. But he proceeded in such secrecy that even his closest acquaintances and colleagues were unaware of the extent of his effort.

Last week, Wiles was finally ready to reveal that he had proved a significant part of the Taniyama conjecture. He chose to describe his results in his native land during three lectures presented at a workshop at the recently opened Isaac Newton Institute for Mathematical Sciences of the University of Cambridge in England, the university where Wiles had done his doctoral studies. His audience included Mazur, Ribet, and many other experts in this particular specialty.

At the end of his third lecture, almost as an afterthought, Wiles noted that he had proved enough of the Taniyama conjecture to show that Fermat's last theorem was true.

"The feeling in the field had been that the Taniyama relationship - this conjecture for certain curves -- was absolutely untouchable, an incredibly deep, very difficult problem to solve," says Princeton's Henri Darmon. "Wiles really shocked the mathematical community by announcing he had proved a large part of the Taniyama conjecture."

By following a course that built on previous, well-understood results, and because of his own reputation for being extremely cautious and careful in his mathematical work, Wiles has already earned a great deal of respect for his proof. Nonetheless, the details of Wiles' 200-page proof need to be checked thoroughly by experts. That might take as long as a year.

"There are a number of subtle points," Darmon says. "But given that he's using fundamental theorems, the basic ideas seem correct."

Five years ago, Yoichi Miyaoka created a considerable stir when he announced that he had proved Fermat's last theorem - using an approach that differed considerably from the one taken by Wiles. However, Miyaoka's proof turned out to be flawed (SN: 4/9/88, p. 230).

If Wiles' proof holds up, it does far more than establish Fermat's conjecture as a theorem. Mathematicians now have new techniques - developed by Wiles - for tackling other important, difficult questions in number theory.

"For the specialist, that he has proved Fermat's conjecture is the less exciting part," Darmon says. "His work completely changes the field."

In fact, says Sarnak, "this is not the end of a subject, but the beginning."