From: John Young <jya@pipeline.com>
Date: Tue, 31 Jan 95 07:09:03 PST
To: cypherpunks@toad.com
Subject: NYT on Fermat Proof
Message-ID: <199501311508.KAA20666@pipe1.pipeline.com>
MIME-Version: 1.0
Content-Type: text/plain


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.