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.