Last word not yet in on Fermat's conjecture.

Last June, mathematician Andrew Wiles of Princeton University stunned the mathematical community and drew worldwide attention when he dramatically announced a proof of Fermat's last theorem. A year later, the arguments presented by Wiles stand as a considerable mathematical achievement, but a troublesome gap in his lengthy chain of reasoning leaves Fermat's last theorem unproved.

"It took people a while to realize how serious the problem was," says Kenneth A. Ribet of the University of California, Berkeley "And it has not yet been taken care of."

Fermat's last theorem asserts 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. A proof of this assertion's validity has eluded mathematicians for more than 350 years.

In his approach, Wiles followed up several key discoveries made by other mathematicians during the 1980s. These insights linked Fermat's last theorem to important ideas in number theory, particularly to mathematical entities known as elliptic curves (SN: 6/20/87, p.397).

Wiles took advantage of these connections in his announced proof of certain cases of the so-called Taniyama-Shimura conjecture, which in turn establishes the truth of Fermat's last theorem. He outlined his reasoning in three lectures last June at the University of Cambridge in England (SN: 7/3/93, p.5).

"His overall strategy looked wonderful," Ribet notes.

Wiles then submitted a preliminary, 200-page manuscript of the proof to the journal Inventiones Mathematicae. In turn, copies of the manuscript were sent out to about half a dozen mathematicians for checking, but no copies were circulated publicly. Wiles himself continued to work quietly on his proof, clarifying arguments and correcting problems pointed out by the referees.

Late last year, Wiles broke his silence and sent out an electronic-mail message acknowledging a gap in what he had thought was an airtight proof (SN: 12/18&25/93, p.406). The problem involved calculating a precise upper limit on the size of a mathematical object called the Selmer group. Without confirming that this group is small, the proof remained incomplete.

Disappointed but intent on continuing, Wiles said at the time, "I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures."

Starting in February, Wiles gave a detailed account of his work in a course he was teaching at Princeton, covering parts of the proof that had already been checked and verified. Meanwhile, he refined the first few chapters of his lengthy manuscript into a form suitable for circulation.

Wiles also expended a great deal of effort thinking about the obstacle lying in his path toward Fermat's last theorem. In a rare public lecture, which took place last month at the Institute for Advanced Study in Princeton, N.J., a relaxed Wiles explained that he now believes the difficulty is probably surmountable -- that he understands the problem well enough to see its connection with something more standard in mathematics.

"I think his intuition is that he's very close and that no new ideas are needed," says Peter C. Sarnak, one of Wiles' Princeton colleagues. "But you never know with something that looks standard. There may be some subtle issue here which makes it nonstandard."

Wiles will continue his lecture course at Princeton this fall, when he will at last reach the point in the proof where his difficulty lies. He may have an answer by then or he may not.

As mathematicians begin to understand the details of what Wiles has accomplished, it's possible that someone else may find a way to bridge the gap while preserving Wiles' original line of reasoning. Alternatively, someone may completely rework the argument, using some of Wiles' ideas in a different way.

"Or [Fermat's last theorem] could be proved from some completely different perspective," Ribet says.

At the same time, despite the gap and the complexity of the approach, Wiles' work has already inspired several mathematical efforts on related questions. "Any time there's a major achievement like this, it changes what people work on and how they think about things," Ribet says.

"Mathematically he's made a major breakthrough in the subject," Sarnak adds.

But the final step toward Fermat's last theorem remains untrodden. Quoted in the June Scientific American, veteran mathematician Andre Weil of the Institute for Advanced Study said, "[T]o some extent, proving Fermat's theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest."

"People are now even more convinced than they were that Fermat's last theorem is true and ought to be proved," Ribet says. "Probably, someone is going to prove it. The question is how long that is going to take."