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IIRC, there was such a result as "there is no more than 1 non-trivial solution of $x^n+y^n=z^n$, if any", wasn't it? (IIRC, Siegel theorem implies that there are finitely many solutions for $n>3$; so it is the "no more than 1" part that is of particular interest).

Also, any reviews of pre-Wiles' results on Fermat's Last Theorem are appreciated.

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    Would that be up to common factors?2012-07-29
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    @HenningMakholm Yes.2012-07-29
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    Faltings' Theorem implies that for $n>3$, there are at most finitely many solutions.2012-07-29
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    @HenningMakholm Yes... modulo IIRC, sorry. Precisely formulated result is the expected answer to my question.2012-07-29
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    @KeenanKidwell What about "there are at most 1 solution"? Yes, I may incorrectly remember... but it seems to me, that I had read such a result. That result was specific to Fermat's Last Theorem, and not to any homogeneous polynomial.2012-07-29
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    The only "at most one solution" result I can imagine in this context is that for given $x,y,z$ there is at most one $n$ such that $x^n+y^n=z^n$ (excluding trivialities like $x=1,y=-1,z=0$).2012-07-30

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