# Brocard’s conjecture

(Henri Brocard) With the exception of 4 and 9, there are always at least four prime numbers^{} between the square of a prime and the square of the next prime. To put it algebraically, given the $n$th prime ${p}_{n}$ (with $n>1$), the inequality $(\pi (p_{n+1}{}^{2})-\pi (p_{n}{}^{2}))>3$ is always true, where $\pi (x)$ is the prime counting function.

For example, between ${2}^{2}$ and ${3}^{2}$ there are only two primes: 5 and 7. But between ${3}^{2}$ and ${5}^{2}$ there are five primes: a prime quadruplet^{} (11, 13, 17, 19) and 23.

This conjecture remains unproven as of 2007. Thanks to computers, brute force searches have shown that the conjecture holds true as high as $n={10}^{4}$.

Title | Brocard’s conjecture |
---|---|

Canonical name | BrocardsConjecture |

Date of creation | 2013-03-22 16:40:53 |

Last modified on | 2013-03-22 16:40:53 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 8 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 11A41 |

Synonym | Brocard conjecture |

Related topic | LegendresConjecture |