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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? – Henning Makholm Jul 29 '12 at 21:45
@HenningMakholm Yes. – some_math Jul 29 '12 at 21:51
Faltings' Theorem implies that for $n>3$, there are at most finitely many solutions. – Keenan Kidwell Jul 29 '12 at 21:59
@HenningMakholm Yes... modulo IIRC, sorry. Precisely formulated result is the expected answer to my question. – some_math Jul 29 '12 at 22:03
@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. – some_math Jul 29 '12 at 22:09

The Wikipedia article gives a good summary of pre-Wiles work. One highlight: In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes $p$.

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