# Legendre’s conjecture

(Adrien-Marie Legendre) There is always a prime number^{} between a square number and the next. To put it algebraically, given an integer $n>0$, there is always a prime $p$ such that $$. Put yet another way, $(\pi ({(n+1)}^{2})-\pi ({n}^{2}))>0$, where $\pi (x)$ is the prime counting function.

This conjecture was considered unprovable when it was listed in Landau’s problems in 1912. Almost a hundred years later, the conjecture remains unproven even as similar conjectures (such as Bertrand’s postulate) have been proven.

But progress has been made. Chen Jingrun proved a slightly weaker version of the conjecture: there is either a prime $$ or a semiprime $$ (where $q$ is a prime unequal to $p$). Thanks to computers, brute force searches have shown that the conjecture holds true as high as $n={10}^{5}$.

Title | Legendre’s conjecture |
---|---|

Canonical name | LegendresConjecture |

Date of creation | 2013-03-22 16:38:17 |

Last modified on | 2013-03-22 16:38:17 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 11A41 |

Related topic | BrocardsConjecture |