Twin primes | (p, p + 2) are both prime. |

Prime triplets | (p, p + 2, p + 6) or (p, p + 4, p + 6) are all prime. |

Prime quadruplets | (p, p + 2, p + 6, p + 8) are all prime. |

Cousin primes | (p, p + 4) are both prime. |

Sexy primes | (p, p + 6) are both prime. |

Form 3n+1 | Of the form . |

Form 4n+1 | Of the form . |

Form 4n+3 | Of the form . |

Emirp | Reversible primes |

Euler prime numbers | Of the form . |

Mersenne primes | Of the form . |

Euclid primes | Of the form . |

Primorial primes | Of the form . |

Fermat primes | Of the form . |

Factorial primes | Of the form n! − 1 or n! + 1 . |

Double factorial primes | Of the form n!! − 1 or n!! + 1 . |

Sophie Germain primes | p and 2p + 1 are both prime . |

Safe primes | p and (p - 1) / 2 are both prime . |

Left-truncatable primes | |

Right-truncatable primes | |

Two-sided primes | |

Wilson primes | p^{2} divides (p − 1)! + 1 . |

Super primes | p-th prime |

Odd primes | Of the form . |