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# Brocard's conjecture

In number theory, **Brocard's conjecture** is a conjecture that there are at least four prime numbers between (*p*_{n})^{2} and (*p*_{n+1})^{2}, for *n* > 1, where *p*_{n} is the *n*^{th} prime number.^{[1]} It is widely believed that this conjecture is true. However, it remains unproven as of March 2015.

n | \( p_n\) | \( p_n^2\) | Prime numbers | \(\Delta \) |
---|---|---|---|---|

1 | 2 | 4 | 5, 7 | 2 |

2 | 3 | 9 | 11, 13, 17, 19, 23 | 5 |

3 | 5 | 25 | 29, 31, 37, 41, 43, 47 | 6 |

4 | 7 | 49 | 53, 59, 61, 67, 71… | 15 |

5 | 11 | 121 | 127, 131, 137, 139, 149… | 9 |

\( \Delta\) stands for \(\pi(p_{n+1}^2) - \pi(p_n^2) \). | ||||

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEIS A050216.

Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for *p*_{n} ≥ 3 since *p*_{n+1} - *p*_{n} ≥ 2.

Notes

Weisstein, Eric W., "Brocard's Conjecture", MathWorld.

See also

Prime counting function

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