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!
Asked
Active
Viewed 795 times
5
-
3The abc conjecture only implies FLT for sufficiently large n. There are rumors that a proof is near completion. – Ragib Zaman Aug 04 '12 at 04:23
-
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). – M Turgeon Aug 08 '12 at 21:43
-
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_consequences – Karatuğ Ozan Bircan Mar 07 '13 at 00:19
1 Answers
5
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.
