Fermat's Theorem
(The article below was written for the Clare Association
Annual in 1994. It describes, in language intended for the
"intelligent layman", the history, the importance and
the proof of Fermat's Theorem.)
Last year saw possibly the most remarkable
developments in mathematics this century - and Clare College was
at the centre of it. For over 350 years mathematicians all over
the world have tried without success to prove what has become
known as ``Fermat's Last Theorem''. But recently a proof was
announced by Andrew Wiles, using the help of Richard Taylor. A.J.
Wiles (1974) was a graduate student, and then Research Fellow, in
the late 1970s and is now a Professor at Princeton University;
R.L. Taylor (1980), who was Wiles' research student, is currently
a Fellow of Clare and Reader in Pure Mathematics.
Does an+bn=cn
have a
solution in integers if a,b,c >
0 and
n > 2 ?
Fermat's theorem is very easy to describe. Most people have
come across the fact that 32+42=52
(most likely during a study of right-angled triangles). Similar
examples, such as 52+122=132,
are not hard to find. Indeed there are infinitely many integer
solutions to the equation a2+b2=c2,
but it is possible to describe all of them in an explicit way.
The 17th century French mathematician Pierre de Fermat thought
about equations with higher powers, such as a3+b3=c3,
a7+b7=c7
and so on. He famously claimed that the equation an+bn=cn
has no solution in non-zero integers n, a, b
and c if n is larger than 2.
However, Fermat gave no proof, writing that `the proof is
wonderful but the margin is too small to contain it'. Ever since,
mathematicians have been trying to verify Fermat's claim.
Finding a proof of this assertion has been the most
tantalizing problem in the history of mathematics. It is the
simplicity of the statement of Fermat's theorem which has made it
so popular amongst even those with only an elementary
mathematical background. Indeed it was as a schoolboy in
Cambridge at the age of only nine that Andrew Wiles (whose
father, Maurice, was Dean of Clare) read about the theorem and
became fascinated by it.
It
doesn't very often.
The importance of Fermat's theorem in mathematics has been not
so much in the theorem itself as in the tremendous theoretical
developments which can be traced to attempts to prove the
theorem. For instance, a large part of modern mathematics has
grown out of the efforts of 19th century mathematicians
(particularly Kummer) to solve the puzzle. In the 20th century
the development and abstraction continued apace. But especially
significant breakthroughs occured in th 1980's. Gerd Faltings
proved that, for any given value of n, there are only
finitely many solutions. In 1986 Ken Ribet proved that Fermat's
theorem would follow if a certain claim about elliptic curves,
the so-called Tanimaya-Shimura-Weil Conjecture, were true.
It never
does!
It was at this point that Wiles decided that proving Fermat's
theorem was a real possibility and no longer just a fantasy.
Elliptic curves have been studied thoroughly by mathematicians
for over a hundred years, and Wiles knew as much about them as
anyone in the world. Over the next seven years he devoted himself
to the task. Then, in June 1993, he announced his proof.
Er, maybe
...
Every mathematician has had the experience of overlooking a
small gap in an argument. This was Wiles' experience here. As
experts the world over (including Richard Taylor's group here in
Cambridge) spent the next few months poring over the proof, small
holes appeared. All of them could be patched up except one, which
unfortunately defied efforts to repair it. Of course, a proof
with a tiny hole in it is no proof at all.
No,
really!
In the attempt to repair the proof a considerable part of the
argument had to be overhauled. The hole involved a complicated
construction called an Euler system, developed by Wiles
specifically for a part of the proof. He had originally attempted
to prove Fermat's theorem using Hecke algebras but had
encountered a stumbling block. But it was then realised that a
different argument using Hecke algebras would work.
By this time Richard Taylor and Andrew Wiles were working
together. Abandoning Euler systems altogether, they were able to
overcome the barrier, and so the proof was back on the tracks (in
fact in a simpler form than originally).
The new proof was announced in the autumn of 1994. Many people
have seen the manuscript (a few can even understand it!) and so
far no-one has raised any objection. So this time it looks as
though the proof really will stand. For mathematicians it is a
remarkable thing that a conjecture we grew up with as being
forever a mystery should now be an established theorem. For those
who specialise in number theory there is delight that their
efforts over many decades have been in the right direction, and
their theory is now richer than ever. The achievement by Andrew
Wiles with Richard Taylor is a great one. Those of us in Clare
may enjoy a peculiar pride that the College is so intimately
bound up in this historical moment.
|