A few hundred years of brilliant minds throwing themselves at it didn't find anything simpler than Wiles' approach, and now that the first proof has been found

A few hundred years of brilliant minds throwing themselves at it didn't find anything simpler than Wiles' approach, and now that the first proof has been found it seems fairly unlikely that any serious science will get expended specifically towards finding another proof of the Theorem. 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/ 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.