Catalan's conjectureCatalansConjecture2008-01-18 15:12:532008-01-27 16:23:46Conjecturenumber theory% this is the default PlanetMath preamble. as your knowledge
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The successive positive integers 8 and 9 are integer powers of positive integers ($2^3$ and $3^2$), with exponents greater than 1. {\em 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.