zlacker

Well, the answer is "simple": It's because the modularity theorem was proven (or better, the Tanyiama-Shimura conjecture was proven)

But why that solves the problem? Because it connects two branches of mathematics (modular forms and elliptic equations) in a way that proves that equations of that form cannot exist (where the exponent is > 2)

Though there probably is an easier way of explaining it, it is strongly suspected that Fermat got the wrong idea there.

>>raverb+(OP)
I still like the idea that Fermat had a legit proof and that one day a simpler one will be found.

I also like that FLT follows easily from the Beal conjecture, which seems overlooked. Maybe its overlooked because its closely related to some other (harder to understand) conjectures.