A long article (16k) today on Dr. Andrew Wiles's dramatic, successful struggle to close the gap in his proof of Fermat's last theorem. For email copy send blank message to <jya@pipeline.com> with subject: FERM_hak Here are excerpts: Fermat's last theorem which has tantalized mathematicians for more than 350 years, has at last been solved, say those who have read the revised but not yet published proof. But the endgame of this furious chase has proved as full of last-minute surprises as a murder mytery. For Dr. Andrew Wiles of Princeton University, the chief author of the proof, triumph had to be snatched from the jaws of disaster. His first proof, which aroused world-wide attention when announced two years ago, turned out to contain a gap, which Dr. Wiles found he was unable to cross alone. *** But he went back to the Flach method for one last time. "There was one variant in the original argument that I'd convinced myself wouldn't work but I hadn't convinced him," Dr. Wiles said. "I was sitting at my desk one morning really trying to pin down why the Flach method wasn't working when, in a flash, I saw that what was making it not work was exactly what would make a method I'd tried three years before work. It was totally unexpected. I didn't quite believe it." He dashed down from the attic to tell his wife. Although his enthusiasm was infectious, Dr. Wiles said, " I actually think she didn't believe me." *** Dr. Wiles said that the breakthrough came in figuring out how to glue together an infinite collection of mathematical objects called Hecke rings. He had initially been creating what was "a very natural relationship between these objects -- natural in the sense that you can give a clear definition of the maps between them." It was an inductive argument. The idea was to take one element of a set and use that to find the next element, then to use the second to construct a third, and so on. The new idea, Dr. Wiles said, "was to simply construct artificial maps between these objects." "You wouldn't show a relationship explicitly," he said, but would use a counting argument to prove that a relationship had to exist. The basic idea is to use the pigeonhole principle: if you have more objects than pigeonholes to put them in, then at least one pigeonhole must contain more than one object. The complete argument involves creating an infinite sequence of sets of pigeonholes and then showing that there must be objects that show up in every set of pigeonholes. This allowed.Dr. Wiles and Dr. Taylor to prove that there must be an infinite set of Hecke rings that share a relationship, although they never have to specify exactly what that relationship is.