Fini To Fermat's Last Theorem

By MICHAEL D. LEMONICK

The mathematicians who gathered in a Cambridge University lecture room last Monday had no idea that they were about to witness history. They had come to hear Andrew Wiles, an English colleague based at Princeton University, give three one-hour lectures on "Modular Forms, Elliptic Curves and Galois Representations," an abstract topic even by the rarefied standards of higher math. By the end of the first hour, though, they knew something was up. Recalls Nigel Boston, a visiting mathematician at Cambridge's Isaac Newton Institute: "We realized where he could be heading. People were giving each other wide-eyed looks." By the end of the third hour, the room was packed with excited number theorists. Wiles finished up his talk and wrote a simple equation on the blackboard, a mathematical afterthought that logically followed from all that he had been saying -- and the audience burst into wild applause.

Wiles had unraveled the greatest unsolved mystery of mathematics. Known as Fermat's Last Theorem, it has baffled number experts for more than 350 years. A handful of solutions have appeared over the centuries -- the latest in 1988 -- and then been retracted upon discovery of a flaw. But, says University of California, Berkeley, mathematician Kenneth Ribet, "Wiles has a first-rate reputation in the subject. He is careful, and he is methodical; he does very, very good work . . . and he presented beautiful arguments." Within an hour, electronic mail hailing the achievement began streaking across the globe to universities and research centers.

What makes the theorem so tantalizing is that for all its fiendish difficulty to prove, it is almost absurdly simple to state. The ancient Greeks knew that the equation x (squared) + y (squared) = z (squared) could be correct if x, y and z were replaced by certain integers -- that is, ordinary nonfractional numbers. For example, 3 (squared) + 4 (squared) (that is, 9 + 16) equals 25, which is 5 (squared). Substituting 5, 12 and 13 for x, y and z works too, and so do other combinations.

In 1637 or so, a French lawyer, poet, classicist and mathematician named Pierre de Fermat declared that such solutions exist only for squares. Raise the exponent to any number higher than 2 -- change the equation to x 7 + y 7 = z 7, for example, or x 12 + y 12 = z 12 -- said Fermat, and no combination of integers will work. "I have found a truly wonderful proof," wrote Fermat in the margin of a book, "which this margin is too small to contain." He lived until 1665 but never did write it down -- evidence, many believe, that he hadn't proved the proposition after all.

Fermat had a sufficiently august reputation, though -- he laid the foundation for probability theory and analytic geometry theory -- that his tantalizing claim lured generations of mathematicians into attacking the problem. They failed, but in the process, says University of Illinois * mathematician Lee Rubel, they "generated an awful lot of extremely important and powerful mathematics -- it has been a seed for major developments." In fact, the mathematical fallout from Fermat's theorem has turned out to be more significant than the original theorem itself. For decades, Fermat's Last Theorem has been a kind of backwater in math, its significance more symbolic than real. It would most probably be solved in the course of addressing some broader problem.

That is just the way a Japanese mathematician, Yoichi Miyaoka, seemed to have cracked the theorem in 1988: he apparently (but wrongly) showed that there was a link between Fermat's Last Theorem and a proven proposition in a field known as differential geometry.

Wiles' solution comes at the theorem in a different way. What he actually proved was an important part of another math puzzle, known in the trade as the Taniyama Conjecture, which deals with the equations that describe mathematical objects known as elliptic curves. Just six years ago, Berkeley's Ribet demonstrated that proving this conjecture was tantamount to proving Fermat's Last Theorem. "What is amazing about Wiles' proof," says Boston, "is that while it built on previous attempts, Andrew realized how to put all these complicated pieces together."

There is always a chance that Wiles, too, has made a mistake. His proof runs more than 200 pages, and he could present only the highlights in the Cambridge lecture. The final test will come in a few months, when Wiles circulates a complete, written version of the proof to others for careful checking. That will not be easy. Says Ribet: "Wiles' arguments are based on the most advanced, most elaborate mathematics that exist in this field. The number of mathematicians who can really fully understand the arguments would fit into a conference room."

Wiles' proof is historic, but the subfields of mathematics generated along the way by people working to solve Fermat's theorem are full of perplexing problems, and so are other areas of math. A proof of Fermat's famous theorem by no means brings any line of inquiry to an end. Still bedeviling mathematicians are the Poincare Conjecture, the Riemann Hypothesis, Goldbach's Conjecture, Kepler's sphere-packing problem and dozens of others. There are, in short, enough mind-bending challenges to keep mathematicians busy for at least the next 350 years.

With reporting by David Bjerklie/New York