# Andrica’s conjecture

Conjecture (Dorin Andrica). Given the $n$th prime ${p}_{n}$, it is always the case that $1>\sqrt{{p}_{n+1}}-\sqrt{{p}_{n}}$.

This conjecture remains unproven as of 2007. The conjecture has been checked up to $n={10}^{5}$ by computers.

The largest known difference of square roots of consecutive primes happens for the small $n=4$, being approximately 0.67087347929081. The difference of the square roots of the primes ${10}^{314}-1929$ and ${10}^{314}+2318$ (which cap a prime gap of 4247 consecutive composite numbers^{} discovered in 1992 by Baugh & O’Hara) is a very small number which is obviously much smaller than 1.

## References

- 1 D. Baugh & F. O’Hara, “Large Prime Gaps” J. Recr. Math. 24 (1992): 186 - 187.

Title | Andrica’s conjecture |
---|---|

Canonical name | AndricasConjecture |

Date of creation | 2013-03-22 16:41:22 |

Last modified on | 2013-03-22 16:41:22 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 8 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 11A41 |

Synonym | Andrica conjecture^{} |