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### 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/
```
Associated Topics:
High School History/Biography
High School Number Theory

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