Fermat's Theorem

Date: 01/21/98 at 01:08:30
From: Vincent M. Li
Subject: Fermat's Theorem

I have heard a lot about Fermat's Theorem, and how it was once 
considered one of the world's greatest unsolved mathematical mystries. 
Supposedly it has been solved by a Professor Andrew Wiles. I was just 
wondering why it was such a mystey, and how it was proved. I probably 
won't understand the implications of this for the known world but I am 
curious to learn.

Date: 01/21/98 at 15:20:02
From: Doctor Wilkinson
Subject: Re: Fermat's Theorem

Here's a webpage with a lot of information:

  http://www.best.com/~cgd/home/flt/flt01.htm   

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

Date: 01/21/98 at 15:40:21
From: Doctor Rob
Subject: Re: Fermat's Theorem

You can try the following URL for some of the history of this problem:

  http://daisy.uwaterloo.ca/~alopez-o/math-faq/node22.html   

Also go to our Archives search form:

  http://mathforum.org/mathgrepform.html   

and search for "Wiles". This will give you a lot of information on 
this subject.

It was such a mystery because Fermat not only stated the problem, but
claimed he had a solution. 350 years passed with many of the greatest
mathematicians straining to rediscover that solution, but failing. It 
is now thought that Fermat was mistaken. Clearly the solution by Wiles 
is not the one Fermat thought he had - he could not have known even a 
fraction of what Wiles used in his proof. It is also a mystery because 
it is so easy to state that any student of beginning algebra can 
understand the problem, but the proof is so extremely difficult.

The first step in learning the kind of math you would need to 
understand Wiles's proof is to learn about curves in the plane. You 
already may be familiar with those of degree 2, the ellipse, parabola, 
and hyperbola. The study of curves of degree 3 is the jumping-off 
point for this subject.

Most such curves are called "elliptic curves" because of their 
connection will elliptic functions and elliptic integrals. Elliptic 
integrals are so named because you need to evaluate one to find the 
arc length of an ellipse. As you know, elliptic integrals cannot be 
expressed in closed form in terms of the usual functions of 
mathematics. That is where the elliptic functions arise. One elliptic 
function, the Weierstrass P-function, and its derivative P' satisfy an 
equation of degree 3, called an elliptic curve.

The particular elliptic curve Wiles used is

   y^2 = x*(x + a^n)*(x - b^n),

where a and b are positive integers, and n is a prime number >= 5.
This is how we get a connection to Fermat's equation a^n + b^n = c^n.

Elliptic curves have many wonderful and surprising properties. For one 
thing, the points on the curve with a certain operation form an 
abelian group. The interplay between the abstract algebra (groups), 
complex analysis (elliptic functions), and geometry (curves) leads to 
some very powerful and exciting mathematics. Elliptic curves seem to 
have applications to a surprisingly diverse set of areas of 
mathematics, such as solving Diophantine equations, sphere packing, 
and proving large integers prime.

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