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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!

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    If 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_consequences2013-03-07

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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.