# Brocard’s problem

Brocard’s problem, first posed by Henri Brocard in 1876, asks for factorials^{} that are one less than a square, that is, solutions to the equation $n!+1={m}^{2}$. Only three solutions are known: $4!+1={5}^{2}$, $5!+1={11}^{2}$ and $7!+1={71}^{2}$. Srinivasa Ramanujan also pondered the problem, in 1913. Erdős believed that there are no other solutions, and no more have been found for $n$ up to ${10}^{9}$.

## References

- 1 P. Erdős, & R. OblÃÂ¡th, “Über diophantische Gleichungen der Form $n!={x}^{p}\pm {y}^{p}$ und $n!\pm m!={x}^{p}$” Acta Szeged. 8 (1937): 241 - 255

Title | Brocard’s problem |
---|---|

Canonical name | BrocardsProblem |

Date of creation | 2013-03-22 18:09:59 |

Last modified on | 2013-03-22 18:09:59 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A25 |