# near-square prime

A near-square prime^{} is a prime number^{} $p$ of the form ${n}^{2}+k$, with $n$ being any integer and $$ also an integer. Since for any nonzero real number $x$ it is always the case that ${x}^{2}\ge 0$, it doesn’t matter if $n$ is negative.

5 | 149 | |||||||||||

4 | 29 | 53 | ||||||||||

3 | 67 | 103 | ||||||||||

2 | 11 | 83 | ||||||||||

1 | 5 | 17 | 37 | 101 | ||||||||

0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |

$-1$ | 3 | |||||||||||

$-2$ | 7 | 23 | 47 | 79 | ||||||||

$-3$ | 97 | |||||||||||

$-4$ | ||||||||||||

$-5$ | 31 | 59 | 139 |

Fermat primes^{} are near-square primes for $k=1$ with the additional requirement that $n={2}^{{2}^{m}-1}$, while Carol primes are near-square primes for $k=-2$ with the additional requirement that $n={2}^{m}-1$.

For $k=-1$, only $n=2$ gives a prime, namely 3.

Title | near-square prime |
---|---|

Canonical name | NearsquarePrime |

Date of creation | 2013-03-22 18:57:37 |

Last modified on | 2013-03-22 18:57:37 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A41 |