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If interested there is a very good book about this subject: http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-P...
I have a question about these hundreds-of-pages long proofs. Who can check such long proofs and be confident about it? To study a document of this size and advanced content rigorously, you would need years - and even then, it is not certain that you didn't overlook something. How can I ever be confident in such proofs until they are checked in, say, Coq or something?
Earlier quoted context omitted.
I don't know if you're being sarcastic, but Wiles had to first prove the Modularity theorem for a certain subset of elliptic curves, which lay the ground work for proving the Modularity theorem in general. This is one of the major achievements of modern mathematics.
"marginal improvement" was the key phrase
I have a question about these hundreds-of-pages long proofs. Who can check such long proofs and be confident about it? To study a document of this size and advanced content rigorously, you would need years - and even then, it is not certain that you didn't overlook something. How can I ever be confident in such proofs until they are checked in, say, Coq or something?
Huh, I thought I had read somewhere that a machine checked proof of Fermat's Last Theorem had already been created. I must have mixed it up with something else.
Well, the margin of Arithmetica is certainly too narrow to contain it. Can any mathematicians confirm that it is truly marvellous? Or is there another proof out there?
Lots of fresh crackpot non-proofs available, though, if you're into those! There are plenty of simply-stated problems in number theory that do not yield to straightforward proof, but Fermat hit marketing gold by saying that he had such a proof already. Who wouldn't want to think that a few years of the common core gives you at least the mathematical skills of a 17th century fancylad?
A common theory about Fermat's claimed "proof" is that he mistakenly assumed that the technique he used to prove the cases for n=3 and n=4 (he called it "infinite descent", basically proof by induction + contradiction) would generalise to higher n. You might broadly think of it like saying "I have a technique for finding prime numbers! You take any integer then double it and add one". Well, you get lucky for a few cases but the machinery isn't there at the back end. Long and short of it is that until some magnificent new areas of the science open up, Wiles' proof is probably the simplest we're likely to get; indeed it doesn't get much simpler than just citing it [Wiles 1995].
BTW there's a pretty great blog at http://fermatslasttheorem.blogspot.co.uk/ which tries to present Wiles' theorem in bite-size chunks, if you're interested in exploring whether you'd regard it as truly marvellous.