TITLE: Theorems which say "such and such method cannot possibly prove FLT"
QUESTION [19 upvotes]: 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.

REPLY [8 votes]: There is also the relative approach, which is to show that making Method X work is at least as difficult as solving some set of known hard problems, due to relatively simple reductions between the problems.  This is what has been done for the P=NP problem --- thousands of NP problems are computationally equivalent to each other --- as well as the Riemann Hypothesis which has a large number of known equivalents.  If you had the bright idea of replacing the non-elementary Riemann zeta function and statements about its zeroes by statements about the distribution of primes, or sums of the Moebius function, then the translation between those contexts is known and much easier than the theory of the zeta function itself, so there would have to be some elementary idea that people had missed in each and every one of those equivalent environments. 
For example, if Method X = "prove an effective form of the ABC conjecture" (the non-effective form implies FLT for large exponents using a few lines of algebra), then solutions of dozens of hard unsolved problems, and easier proofs of very hard theorems, would be assured as soon as one could carry out Method X.  Because the reductions of these problems to the ABC conjecture are often quite simple, any elementary idea used to carry forward Method X would immediately translate into elementary methods for the dozens of other problems, and it is less likely that nobody would have ever noticed this in any of the other actively studied problems.