Word: theorems
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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 A handful of solutions have appeared over the centuries -- the latest in 1988 -- and then been retracted upon discovery of a flaw.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...
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...
...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...