# Catalan’s conjecture

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

$$x=n=3,y=m=2.$$ |

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. 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 ${x}^{y}={y}^{x}$ (http://planetmath.org/solutionsofxyyx).

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 |