# 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

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

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.

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