Catalan’s conjecture

The successive positive integers 8 and 9 are integer powers of positive integers ($2^3$ and $3^2$), with exponents greater than 1. Catalan’s conjecture (1844) said that there are no other such successive positive integers, i.e. that the only integer solution of the Diophantine equation

$$x^m-y^n=1$$

with  $x>1$,  $y>1$,  $m>1$,  $n>1$  is

$$x=n=3,y=m=2.$$

It took more than 150 years before the conjecture was proven. Mihailescu gave in 2002 a proof in which he used the theory of cyclotomic fields and Galois modules.

For details, see e.g.http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdfthis article.

See also a related problem concerning the equation$x^y=y^x$ (http://planetmath.org/solutionsofxyyx).