# Goormaghtigh conjecture

For integers $x,y,m$ and $n$ satisfying $x>1$, $y>1$, and $n>m>2$, the equation

$$\frac{{x}^{m}-1}{x-1}=\frac{{y}^{n}-1}{y-1}$$ |

has only the solutions $(x,y,m,n)=(5,2,3,5)$ and $(90,2,3,13)$.

See the following paper for some progressâ€™ on the conjecture:

M. Le, Exceptional solutions of the exponential Diophantine equation^{}
$({x}^{3}-1)/(x-1)=({y}^{n}-1)/(y-1)$. J. Reine Angew. Math. 543 (2002) 187-192.

See Section 7 of following paper in which the solution of a certain case of the
conjecture (given in the
latter qouted paper) is used to solve a
problem in group theory concerning the non-cyclic graph of a group. Also see
Proposition^{} 7.6 of
the folloiwng paper for an slightly special case of the conjecture which its
solotion has some
applications in group theory.

Alireza Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Communications in Algebra, 35 (2007) 2057-2081.

Title | Goormaghtigh conjecture |
---|---|

Canonical name | GoormaghtighConjecture |

Date of creation | 2013-03-22 15:18:55 |

Last modified on | 2013-03-22 15:18:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Conjecture |

Classification | msc 11J86 |

Classification | msc 11J61 |