# 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.

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).

Title Catalan’s conjecture CatalansConjecture 2014-12-16 16:16:07 2014-12-16 16:16:07 pahio (2872) pahio (2872) 8 pahio (2872) Conjecture msc 11D45 msc 11D61 Mihailescu’s theorem FermatsLastTheorem SolutionsOfXyYx