Catalan’s conjecture

The successive positive integers 8 and 9 are integer powers of positive integers (23 and 32), 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 equationMathworldPlanetmath


with  x>1,  y>1,  m>1,  n>1  is


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 fieldsMathworldPlanetmath and Galois modules.

For details, see e.g. article.

See also a related problem concerning the equation xy=yx (

Title Catalan’s conjecture
Canonical name CatalansConjecture
Date of creation 2014-12-16 16:16:07
Last modified on 2014-12-16 16:16:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Conjecture
Classification msc 11D45
Classification msc 11D61
Synonym Mihailescu’s theorem
Related topic FermatsLastTheorem
Related topic SolutionsOfXyYx