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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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    The abc conjecture only implies FLT for sufficiently large n. There are rumors that a proof is near completion.2012-08-04
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    For a reference to the result mentionned by Ragib, see Lang's *Algebra*. Also, assuming ABC, we can prove a weak form of FLT, namely, that there exists only finitely many primitive solutions (this also follows from Falting's theorem).2012-08-08
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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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