Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common factors. Fermat's equation has holes, so the Mordell conjecture implies that for each integer , the Fermat equation has at most a finite number of solutions. This conjecture was proved by Faltings (1984).

Elkies, N. D. "ABC Implies Mordell." Internat. Math. Res. Not. 7, 99-109, 1991.

Faltings, G. "Die Vermutungen von Tate und Mordell." Jahresber. Deutsch. Math.-Verein 86, 1-13, 1984.

Ireland, K. and Rosen, M. "The Mordell Conjecture." §20.3 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 340-342, 1990.

van Frankenhuysen, M. "The ABC Conjecture Implies Roth's Theorem and Mordell's Conjecture." Mat. Contemp. 16, 45-72, 1999.

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Weisstein, Eric W. "Mordell Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MordellConjecture.html