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 here. 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 Fermat-Wiles
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