It can easily be proven assuming Szpiro's conjecture that Fermat's Last Theorem is true for sufficiently large $n$. The proof consists of extremely straightforward computations. My question is, is there a refinement of that proof that prove's FLT for all $n$? Or maybe, are there any other unproven conjectures (ABC conjecture, etc.) that if proven would lead to a simple proof of FLT? If so, what is the statement of the conjecture and what is the proof of FLT assuming this conjecture? Thanks!
Proving Fermat's Last Theorem (easily) using "assumed" conjectures
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0If abc conjecture implies Fermat's Last Theorem, then equivalent statements imply as well. See the list on Wikipedia: http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequences – 2013-03-07
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I think that an effective form of ABC will give an effective upper bound on $n$ for which $x^n + y^n = z^n$ has non-trivial solutions. This would leave finitely many $n$ to check, which one could imagine (if the bound on $n$ is not too large) could be checked by other, more traditional means.