Fermat proof flaw: fixing the details.The process of smoothing out and filling in the technical details of the celebrated proof of Fermat's last theorem Fermat's last theorem Statement that there are no natural numbers x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2. , announced last June, has turned up a gap in the proof's logic. Earlier this month, Andrew Wiles For the French mathematician with work in the area of elliptic curves, see . Sir Andrew John Wiles (born April 11 1953) is a British-American research mathematician at Princeton University, specialising in number theory. He is most famous for proving Fermat's Last Theorem. of Princeton University Princeton University, at Princeton, N.J.; coeducational; chartered 1746, opened 1747, rechartered 1748, called the College of New Jersey until 1896. Schools and Research Facilities admitted in an electronic message to colleagues that reviewers of the proof had pointed out a number of problems, one of which remains unresolved. Nonetheless, he noted, "I believe I will be able to finish this in the near future." "Many people have assumed that because the verification hasn't come quickly, there's actually a hole in the proof," says Andrew J. Granville of the University of Georgia Organization The President of the University of Georgia (as of 2007, Michael F. Adams) is the head administrator and is appointed and overseen by the Georgia Board of Regents. in Athens. But "it's not a hole so much as something that needs filling in." Fermat's last theorem asserts that for any whole number n greater than 2, the equation [x.sup.an unspecified amount] + [y.sup.unspecified amount] = z.sup.unspecified amount] has no solution for which x, y, and z are all whole numbers greater than zero. Despite the assertion's simplicity, proof of its validity eluded mathematicians for more than 350 years - until Wiles wile n. 1. A stratagem or trick intended to deceive or ensnare. 2. A disarming or seductive manner, device, or procedure: the wiles of a skilled negotiator. 3. Trickery; cunning. followed up several discoveries made by other mathematicians during the 1980s. These insights linked Fermat's last theorem to important ideas in number theory. Wiles took advantage of these links in his announced proof of part of the socalled Taniyama-Shimura conjecture, which in turn establishes the truth of Fermat's last theorem (SN: 7/3/93, p.5). But filling in the technical details of the proof is a matter of some delicacy, The snag that Wiles has encountered involves 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 remains incomplete. "I am still optimistic op·ti·mist n. 1. One who usually expects a favorable outcome. 2. A believer in philosophical optimism. op that the problems will be worked out," says Karl Rubin Rubin is one of only about half a dozen mathematicians who have copies of the preliminary, 200-page manuscript of the proof. Some mathematicians have complained that Wiles' reluctance to circulate additional copies until his work is finished has hindered the checking process and spawned rampant speculation about where things stand. Wiles plans to present a full account of his work in a series of lectures at Princeton starting in February, |
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