The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite
many solutions in Gaussian integers with no common
factors (Mordell 1922). 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) and hence is now also known as Falting's theorem.
Bombieri, E. "The Mordell Conjecture Revisited." Ann. Scuola Norm. Sup. Pisa Cl. Sci.17, 615-640, 1990.Cornell,
G. and Silverman, J. H. Arithmetic
Geometry. New York: Springer, 1986.Elkies, N. D. "ABC
Implies Mordell." Internat. Math. Res. Not.7, 99-109, 1991.Faltings,
G. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern."
Invent. Math.73, 349-366, 1983.Faltings, G. "Die
Vermutungen von Tate und Mordell." Jahresber. Deutsch. Math.-Verein86,
1-13, 1984.Hindry, M. and Silverman, J. H. Diophantine
Geometry. New York: Springer, 2000.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.Mordell, L. J. "On the Rational Solutions
of the Indeterminate Equation of the Third and Fourth Degrees." Proc. Cambridge
Philos. Soc.21, 179-192, 1922.van Frankenhuysen, M. "The
ABC Conjecture Implies Roth's Theorem and Mordell's Conjecture." Mat. Contemp.16,
45-72, 1999.
holes, so the Mordell
conjecture implies that for each integer 
, the Fermat equation
has at most a finite number of solutions.
