Proof of Fermat's Last Theorem

Date: 12/10/96 at 03:34:57
From: AL_RASHED FAHAD SALEH
Subject: Fermat's last theorem

Dear Dr.Math,

I heard that Fermat's Last Theorem has been proved.  Is that so?  
Can you tell me how it was proved?

Date: 12/10/96 at 11:30:34
From: Doctor Wilkinson
Subject: Re: Fermat's last theorem

Yes, it has been proved.  The proof was published in May of 1995 in 
the Annals Of Mathematics (filling up more than 140 pages!).  The 
proof is based on the theory of elliptic curves and involves a lot of 
high-powered mathematics.  

Let me start by saying that I am not at all qualified to understand
the proof of Fermat's Last Theorem.  What I am telling you is just
what I know from reading articles and watching a videotaped lecture
about the proof.

The theorem states that a^n + b^n = c^n has no solutions in nonzero 
integers if n is an integer greater than 2.

Fermat himself definitely proved the case n = 4, and Euler proved
the case n = 3.  (His proof had errors, but was basically correct).

If a^n + b^n = c^n has no solutions, the same is true for
a^mn + b^mn = c^mn, so it is sufficient to prove the theorem when
n is an odd prime greater than or equal to 5.  This is assumed in the 
Wiles-Taylor proof (the one from May 1995).

So we suppose that l is an odd prime >= 5.  The proof revolves around
the equation:

        y^2 = x (x + a^l)(x - b^l)

considered to be the equation of a curve in the x-y plane.

A curve described by an equation of the form:

        y^2 = x^3 + Ax^2 + Bx + C

with integer coefficients is called an elliptic curve.

Some elliptic curves are called modular.  Being modular means that
there is a particular kind of equation for the number of solutions
of the equation:

        y^2 = x^3 + Ax^2 + Bx + C  (mod p)

for every prime p.

A conjecture called the Taniyama-Shimura-Weil conjecture states that
every elliptic curve with certain restrictions is modular, but until
Andrew Wiles started his work, nobody had ever been able to prove that 
any infinite class of elliptic curves was modular.

The connection with Fermat's Last Theorem was made by two 
mathematicians name Gerhard Frey and Kenneth Ribet.  Frey conjectured 
and Ribet proved that if a^l + b^l = c^l is a counterexample to 
Fermat's Last Theorem, then the elliptic curve 
y^2 = x (x + a^l) (x - b^l) cannot be modular.

Wiles succeeded in proving, on the other hand, that the same elliptic
curve would have to be modular.

This is a contradiction, so the starting assumption that 
a^l + b^l = c^l has to be false.

Wiles first announced that he had a proof in June of 1993.  In the
succeeding months, however, it became clear that there was a serious
gap in the proof.  It took about a year and a half for Wiles and
his student Richard Taylor to find a way around this problem.

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

Date: 12/16/96 at 17:00:50
From: Doctor Ceeks
Subject: Re: Fermat's last theorem

Hi,

I just want to add that in fall of '97 there will be a PBS special
on Fermat's Last Theorem featuring interviews with many of the key
players in the proof.

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