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From: [email protected] (Alex Lopez-Ortiz)
Subject: sci.math FAQ: Status of FLT
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Archive-Name: sci-math-faq/FLT/status
Last-modified: December 8, 1994
Version: 6.2






What is the current status of FLT?







  Andrew Wiles, a researcher at Princeton, claims to have found a proof.
  The proof was presented in Cambridge, UK during a three day seminar to
  an audience which included some of the leading experts in the field.
  The proof was found to be wanting. In summer 1994, Prof. Wiles
  acknowledged that a gap existed. On October 25th, 1994, Prof. Andrew
  Wiles released two preprints, Modular elliptic curves and Fermat's
  Last Theorem, by Andrew Wiles, and Ring theoretic properties of
  certain Hecke algebras, by Richard Taylor and Andrew Wiles.

  The first one (long) announces a proof of, among other things,
  Fermat's Last Theorem, relying on the second one (short) for one
  crucial step.

  The argument described by Wiles in his Cambridge lectures had a
  serious gap, namely the construction of an Euler system. After trying
  unsuccessfully to repair that construction, Wiles went back to a
  different approach he had tried earlier but abandoned in favor of the
  Euler system idea. He was able to complete his proof, under the
  hypothesis that certain Hecke algebras are local complete
  intersections. This and the rest of the ideas described in Wiles'
  Cambridge lectures are written up in the first manuscript. Jointly,
  Taylor and Wiles establish the necessary property of the Hecke
  algebras in the second paper.

  The new approach turns out to be significantly simpler and shorter
  than the original one, because of the removal of the Euler system. (In
  fact, after seeing these manuscripts Faltings has apparently come up
  with a further significant simplification of that part of the
  argument.)

  The preprints were submitted to The Annals of Mathematics. According
  to the New York Times the new proof has been vetted by four
  researchers already, who have found no mistake.

  In summary:

  Both manuscripts have been accepted for publication, according to
  Taylor. Hundreds of people have a preprint. Faltings has simplified
  the argument already. Diamond has generalised it. People can read it.
  The immensely complicated geometry has mostly been replaced by simpler
  algebra. The proof is now generally accepted. There was a gap in this
  second proof as well, but it has been filled since October.

  You may also peruse the AMS site on Fermat's Last Theorem at:


gopher://e-math.ams.org/11/lists/fermat




    _________________________________________________________________



   [email protected]
   Tue Apr 04 17:26:57 EDT 1995

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