The set of prime numbers is a subset of natural numbers that comprehend every element of that set greater than 1 that has no positive divisor other than 1 and itself.

Prime numbers under 100 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

The fundamental theorem of arithmetic states that every natural number greater than 1 can always be presented as the product of prime numbers, and that this product (factorization) is unique.

Every prime number, except 2, is odd. The only consecutive prime numbers are 2 and 3. The consecutive odd prime numbers, those that 2 numbers away, are called twin prime numbers.

#### Prime numbers under 1000.

2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 |

31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 |

73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 |

127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 |

179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 |

233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |

283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 |

353 | 359 | 367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 |

419 | 421 | 431 | 433 | 439 | 443 | 449 | 457 | 461 | 463 |

467 | 479 | 487 | 491 | 499 | 503 | 509 | 521 | 523 | 541 |

547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 | 599 | 601 |

607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |

661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 |

739 | 743 | 751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 |

811 | 821 | 823 | 827 | 829 | 839 | 853 | 857 | 859 | 863 |

877 | 881 | 883 | 887 | 907 | 911 | 919 | 929 | 937 | 941 |

947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 |

#### Twin prime numbers.

Two prime numbers are twin prime numbers when they are 2 numbers away from each other.

#### Pairs of twin prime numbers under 1000.

(3, 5) | (5, 7) | (11, 13) | (17, 19) | (29, 31) | (41, 43) |

(59, 61) | (71, 73) | (101, 103) | (107, 109) | (137, 139) | (149, 151) |

(179, 181) | (191, 193) | (197, 199) | (227, 229) | (239, 241) | (269, 271) |

(281, 283) | (311, 313) | (347, 349) | (419, 421) | (431, 433) | (461, 463) |

(521, 523) | (569, 571) | (599, 601) | (617, 619) | (641, 643) | (659, 661) |

(809, 811) | (821, 823) | (827, 829) | (857, 859) | (881, 883) |

#### Mersenne primes.

A prime number is a Mersenne prime if when being added 1 the result is a power of two. For example, 7 is a Mersenne prime because (7+1=8=2³).

They are named after the philosopher Marin Mersenne (17th century), who stated a series of postulates about them that could only be polished three centuries later.

The first eight Mersenne primes are:

3, 7, 31, 127, 8191, 131071, 524287, 2147483647.

Nowadays we only know 44 Mersenne numbers —the last one was discovered on September 4, 2006.

#### Sophie Germain prime.

A prime number is a Sophie Germain prime if when multiplied by 2 and added 1 the result is also prime. Example: 2 is a Sophie Germain prime because it is a prime number and 2x2+1=5 being 5 a prime too.

#### Sophie Germain primes under 1000.

2 | 3 | 5 | 11 | 23 | 29 | 41 | 53 | 83 | 89 |

113 | 131 | 173 | 179 | 191 | 233 | 239 | 251 | 281 | 293 |

359 | 419 | 431 | 443 | 491 | 509 | 593 | 641 | 653 | 659 |

683 | 719 | 743 | 761 | 809 | 911 | 953 |

#### Fermat number.

A Fermat number is a prime number that makes the following equation true:

where n is a natural number.

We only know five Fermat numbers:

3 (n=0), 5 (n=1), 17 (n=2), 257 (n=3) and 65537 (n=4).