Pure mathematics is a sucker’s game. It lures the curious and confident with its seeming simplicity only to make them look like fools. Consider the equation 9 + 16 = 25. It can be written 3 2 + 4 2 = 5 2. More generally, one can write that a number squared (multiplied by itself) plus a second number squared equals a third number squared. Now the inquisitive are hooked like rubes in three-card monte. Try to find three whole numbers that fit the equation X3 + y3 = Z3. That is, pick numbers for x and y, multiply each by itself twice (like 3 x 3 x 3), and find a third number z which, when multiplied by itself twice, equals the sum of the x3 and y3. There is no such z. That’s what Fermat’s Last states. Nor are there any numbers that fit the equation where the exponents are anything greater than 2. That’s what Fermat claimed to have proved.
Fermat did, in fact, prove his assertion for exponents of 4. Leonhard Euler, the great Swiss mathematician, proved it for exponents of 3 in the 1790s. France’s Adrien Legendre proved it for exponents of 5 in 1823. A few years ago a computer proved it for everything less than 30,000. Things were looking pretty good for Fermat’s Last, but none of this constituted a persuasive proof. What if the number right after the last one checked turned out to falsify the theorem? Only a rigorous proof would do. Their inability to find one, especially in the face of Fermat’s taunt, drove mathematicians crazy.
_B_Too shy:b Wiles approached Fermat’s Last Theorem the way one would a skittish horse obliquely. He started with a 1984 finding that if there are any numbers for which Fermat’s equation holds, then the solutions can be fashioned into something called an elliptic curve. Then Wiles noted a 1987 proof by Ken Ribet of the University of California, Berkeley, that any such elliptic curves could not be of a certain type. (Those who can’t balance their checkbooks can drop out here.) When Wiles proved the contrary-that the relevant elliptic curves are of this type-he had shown the “if” he started with to be wrong: there are no numbers that make Fermat’s equation work. Just as the Frenchman said 356 years ago. This all took Wiles 200 pages. Wiles (who describes himself as too shy to talk to the press) combined ideas from number theory, topology and other disparate fields and basically “threw the kitchen sink” at Fermat’s Last, says Kochen. “Among theorists there is often a sense that something just looks right. It’s the general feeling that [Wiles’s proof] looks right,” he says.
Mathematicians will not know for sure until they check every line, a process that could take years. Thousands of other claims to have proved Fermat’s Last Theorem have fizzled. But if Wiles has triumphed over the historical dare, his proof would promise a huge advance in number theory. It is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato’s shadows, conceal magical laws and orders that the human mind can discover after all.
