Some comments ask whether Pierre de Fermat may have had a proof of his conjecture (which I now call the Fermat-Wiles Theorem) using the mathematics of his day.

Some comments ask whether Pierre de Fermat may have had a proof of his conjecture (which I now call the Fermat-Wiles Theorem) using the mathematics of his day. Agreeing with Colin, on historical grounds, I think Fermat did NOT have a proof of his famous conjecture. The historical reasoning works like this. In Fermat's day, mathematical "publishing" was not submission of articles to peer-reviewed journals, but people interested in mathematics sending personal letters to one another. (One of the first mathematical journals, if I remember correctly, began as a collection of mathematicians' letters to one another, something like turning individual emails into an email list.) Fermat was a lawyer by occupation, and did mathematics in his free time. He corresponded with other mathematicians of his era, including Pascal. In all the surviving correspondence Fermat had with other mathematicians, there is no trace of a proof of "Fermat's Last Theorem," even though, as Colin pointed out, he wrote about simpler cases of that theorem. The real historical reasoning comes in by considering dates of the primary sources. Fermat's note to himself in the margin of the printed Latin translation of Diophantus was written EARLY in Fermat's investigation of number theory. He read that book, taking notes in the margins as he read, and then he wrote for years afterward about interesting number theory problems. The book with Fermat's annotations was published after Fermat's death by Fermat's son, and that is when the marginal note became known to the world, but Fermat had plenty of time--and plenty of pieces of paper with wider margins--to write out a complete proof of his conjecture if he actually had one. So as essentially all mathematicians have concluded on mathematical grounds, and as all historians have concluded on historical grounds, most likely Fermat briefly thought he had a complete proof of his conjecture, which he then worked on, discovering a mistake in his reasoning. He proved a simpler case of the theorem, died, and then when his notes were published, other mathematicians worked on the full conjecture for 350 years before Wiles found the complete proof, based on advanced research by other mathematicians still alive today. Google Books links to the historical background from mathematicians who write good popular books about mathematics: http://books.google.com/books?id=I-RSVN6TjXsC&pg=PT95 http://books.google.com/books?id=I-RSVN6TjXsC&pg=PT95 http://books.google.com/books?id=ZhK73qVgOw0C&pg=PA265 http://books.google.com/books?id=ZhK73qVgOw0C&pg=PA265