The Mathematics of Fermat's Last Theorem
Welcome to one of the most fascinating areas of mathematics.
There's a fair amount of work involved in understanding even approximately
how the recent proof of this theorem was done, but if you enjoy
mathematics, you should find the effort very rewarding.
Please let me know by
email how you like these pages. Let me know if you find any
errors or passages which can readily be improved (short of duplicating
what's in the references listed below).
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Another request I receive frequently is for even more detailed information
about Wiles' proof. The best reference, of course, is Wiles' own paper,
which can be found in the
Annals of Mathematics 141(3), May 1995.
The full text of the article, in PDF format, can be found
Let it suffice to say, however, this is very difficult reading.
So, I will suggest some recent books which provide more background.
If you follow the links, you will be able to purchase the books online.
Simon Singh - Fermat's Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem
- This book is a "popular account", and the author is a
science journalist. It's a good book for the history of the
problem, including the story of how Andrew Wiles eventually
solved it, but most of the mathematics it contains, which is
very little, is in appendices.
One reader says that the book "manages to talk about
mathematics in a way that actually avoids formulas." Please realize
that you will not really understand much if you avoid formulas.
Nevertheless, it's probably the best place to start if you are just
beginning to learn about the subject.
Paulo Ribenboim - Fermat's Last Theorem for Amateurs
- Ribenboim has written more than a few books on number theory, and
several on Fermat's Last Theorem in particular. This one is the most
recent, and probably the best of the lot to learn some of the actual
number theory related to the problem. Anyone with a good foundation
in high school mathematics (without calculus) should be able to
follow most of it, though some of the concepts are sophisticated.
There are many useful references. If you think you have found
an "elementary" proof of the theorem, read this book to help
figure out where you've gone wrong. There are, however, only about
10 pages or so dealing with Wiles' proof itself.
Alf van der Poorten - Notes on Fermat's Last Theorem
- Van der Poorten's book seems to have been the first post-Wiles
book-length technical report on the subject. It has a lot of
valuable background information, both historical and mathematical.
It does contain serious technical mathematics.
But details on Wiles' proof itself are very sketchy.
Yves Hellegouarch - Invitation to the Mathematics of
- Hellegouarch has himself worked on just about all of the
mathematics relevant to the problem, and his book is easily the
best presentation of what is required to understand Wiles' proof,
though it doesn't go much into the actual proof itself. It's
accessible to students who have mastered a few courses of
college/university mathematics. If you have the background, this
is definitely the book to get.
Gary Cornell, Joseph Silverman, Glenn Stevens - Modular Forms and
Fermat's Last Theorem
- This book is the definitive reference on the subject. It
consists of 21 chapters, written by world-class experts on the
subject (including Frey and Ribet, though not Wiles himself). The
chapters are based on a 1995 conference whose purpose was to
present the whole proof, with sufficient supporting details for
math graduate students and professional mathematicians. You should
read and understand Hellegouarch's book before you tackle this one.
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Last updated: September 16, 2003