Proving Fermat's Theorem

Date: 22 Jun 1995 00:15:01 -0400
From: Dionisio Laeber Fleitas
Subject: Fermat's Last Theorem

Has Fermat's Last Theorem already been proved ?
If so, send me the idea of the proof, please.

Date: 22 Jun 1995 09:13:31 -0400
From: Dr. Ken
Subject: Re: Fermat's last theorem

Hello there!

Yes, most people accept that Fermat's Last Theorem has been proven.  
I say "most" because there are only a handful of people in the world 
who can even understand the whole proof (it's huge!), so it's hard to 
say whether all steps are completely rigorous.  The general feeling is 
that the proof is complete.

Here's something that was in the Frequently Asked Questions file for the
newsgroup sci.math:

http://daisy.uwaterloo.ca/~alopez-o/math-faq/node1.html   

Fermat's Last Theorem
  History of Fermat's Last Theorem
 *What is the current status of FLT?
  Wiles' line of attack
  If not, then what?
  Did Fermat prove this theorem?


*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   

Where you'll find documents entitled:

     Network Message Reporting the Wiles Announcement (Ribet)
     Fermat's Last Theorem Discussion (LISTSERV)
     News Item in Notices of the AMS (Ribet)
     Introduction to Fermat's Last Theorem (Cox, Amherst)
     Did Fermat Prove His Last Theorem? (Granville)
     History of Fermat's Last Theorem (Granville)
     Parody on Fermat's Last Theorem (Zorn)
     Fermat's Last Theorem (Ribet, NSF Lecture)
     Fermat's Last Theorem (Mazur, Vancouver Meeting)
     Lecture Notes on Fermat's Last Theorem (Alf van der Poorten)
     Curiousity Gets the Best of Me (Rodgers)
     Some Questions ...... and Some Answers (Adler and Ribet)
     Four Lectures on Fermat's Last Theorem (Allan Adler)
     Wiles Proof of FLT (Rubin and Silverberg, Ohio State)
     SLIDES from Ken Ribet's talk (Cincinnati Meeting, January 1994)
     Published Sources on the History of the Modularity Conjecture

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/