Mordell Conjecture

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 $(n-1)(n-2)/2$ Holes, so the Mordell conjecture implies that for each Integer $n\geq 3$, the Fermat Equation has at most a finite number of solutions. This conjecture was proved by Faltings (1984).

See also Fermat Equation, Fermat's Last Theorem, Safarevich Conjecture, Shimura-Taniyama Conjecture

References

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.