The successive positive integers 8 and 9 are integer powers of positive integers ( and ), 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
with , , , 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 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 (http://planetmath.org/solutionsofxyyx).
|Date of creation||2014-12-16 16:16:07|
|Last modified on||2014-12-16 16:16:07|
|Last modified by||pahio (2872)|