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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. 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. this article.

See also a related problem concerning the equation $x^{y}=y^{x}$.

Related:

FermatsLastTheorem, SolutionsOfXyYx

Synonym:

Mihailescu's theorem

Type of Math Object:

Conjecture

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11D45*no label found*11D61

*no label found*

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