# Puzzles, situation puzzles and recreational mathematics. ## Neuron juice.

The important thing is not to stop questioning. (Albert Einstein.) 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).