PlanetMath: Catalan's conjecture

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Catalan's conjecture

(Conjecture)

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

$\displaystyle x^m-y^n = 1$

with

$\displaystyle x = n = 3, \;\; y = m = 2.$

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.

"Catalan's conjecture" is owned by pahio.

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See Also: Fermat's last theorem

Other names:

Mihailescu's theorem

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Cross-references: modules, cyclotomic fields, theory, proof, conjecture, Diophantine equation, solution, exponents, integers, positive
There are 3 references to this entry.

This is version 2 of Catalan's conjecture, born on 2008-01-18, modified 2008-01-27.
Object id is 10198, canonical name is CatalansConjecture.
Accessed 670 times total.

Classification:

AMS MSC:	11D45 (Number theory :: Diophantine equations :: Counting solutions of Diophantine equations)
	11D61 (Number theory :: Diophantine equations :: Exponential equations)

Pending Errata and Addenda

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