After a previous question that I asked <https://mathoverflow.net/questions/31565/request-for-comments-about-a-claimed-simple-proof-of-flt-closed> was closed, someone suggested in the comments that I ask another question that is more suited for MO. That question is as follows:


Are there any nontrivial theorems of the form "Method X cannot possibly prove FLT."? 


The reason I am asking is because I'd like to know (by deductive reasoning and not relying on 350 years experience) if it is impossible to prove FLT using elementary methods.



Theorems which say "such and such method cannot possibly prove FLT"
For a while at the end of undergrad and beginning of graduate school I made some money correcting an enthusiastic amateur mathematician's incorrect proofs of FLT. (It was a good experience, he was an academic in another field so was professional and was willing to pay what my time was worth.) When I first read his argument I tried to convince him why the method was doomed to failure. He wasn't interested in trying to understand that, but maybe it'll still be informative here.


His method of proof was to consider the "even-ness" (that is the highest power of 2 which is a factor) each side of the equation. He would then do ever more esoteric changes of variables until he inevitably misapplied the tricky part of the non-archimedian triangle inequality (i.e. if 4 divides x and 4 divides y then you have no idea what the "even-ness" is of x+y). At that point he got a contradiction.


Now why is this doomed to failure? Well any reasonable argument along those lines would apply to the 2-adic integers. However, if p is an odd prime, then the power series for the pth root converges 2-adically. So *every* 2-adic integer is a pth power, so certainly the Fermat equation has loads of nontrivial 2-adic integer solutions.


