### The Great Internet Mersenne Prime Search

GIMPS, the “Great Internet Mersenne Prime Search”, has found sixteen Mersenne primes so far. The most recent (and largest) one is 277,232,917 - 1, found on December 26, 2017. See their history page for a list of primes found by GIMPS, and links to an announcement about each one.

For information on GIMPS, including how you can join the search, see the GIMPS home page.

A prime number is a number greater than 1, that is divisible only by itself and 1. The first few prime numbers are: 2, 3, 5, 7, and 11.

A Mersenne Prime is a prime number that is one less than a power of two. That is, a number of the form: 2n - 1, for some number “n”. This can also be written as 2^n - 1, for browsers that do not display subscripts and superscripts.

Mersenne primes are named after Marin Mersenne (1588-1648), a French monk and mathematician.

It is easier (i.e., it takes less computation) to determine whether a Mersenne number is prime, than to test other numbers of the same size. For this reason, the largest known prime is usually a Mersenne prime.

The first few Mersenne primes are:
2 2 - 1 = 2 * 2 - 1 = 3
2 3 - 1 = 2 * 2 * 2 - 1 = 7
2 5 - 1 = 2 * 2 * 2 * 2 * 2 - 1 = 31
2 7 - 1 = 2 * 2 * 2 * 2 * 2 * 2 * 2 - 1 = 127

The next six Mersenne primes correspond to the following values for the exponent “n”: 13, 17, 19, 31, 61, and 89.

In order for 2n - 1 to be prime, the exponent “n” must also be prime. However, if “n” is prime, this does not guarantee that 2n - 1 is prime. For example, 11 is a prime number, but 211 - 1 = 2047 is not prime (and therefore not a Mersenne prime), because 2047 = 23 * 89.

A list of all known Mersenne primes (only 50 of them) is available here.

### Fermat Numbers

Mersenne is only one of many mathematicians with numbers named after him. Pierre de Fermat (1601-1665), who is famous for his “Last Theorem”, also has some numbers named after him. A Fermat number is a number of the form 2(2^n) + 1. Or 2^(2^n) + 1. Fermat knew that the first five numbers (for “n” = 0 to 4) of this form are prime, and he believed that ALL numbers of this form are prime. However, these five Fermat numbers are still the only ones that are known to be prime, and many are known to be composite.

``` n   2^n     2^(2^n) + 1
--------------------------------------
0    1        2 + 1 = 3 = PRIME
1    2        4 + 1 = 5 = PRIME
2    4       16 + 1 = 17 = PRIME
3    8      256 + 1 = 257 = PRIME
4   16    65536 + 1 = 65537 = PRIME

5   32     2^32 + 1 = 4294967297 = 641 * 6700417
6   64     2^64 + 1 = 18446744073709551617 = 274177 * 67280421310721
```
The Fermat numbers F5 thru F32 are all composite (not prime). F14 was proven composite in 1963, but the first factor was not found until 2010. Currently, no factors are known for F20 and F24, though these numbers are known to be composite. At least one prime factor is known for the rest of the Fermat numbers in this range. All of the prime factors are known for F5 through F11.

In 1999, Fermat number F24, which is 216777216 + 1, and has more than 5,000,000 digits, was proven to be composite using Pepin’s test.

Until recently, F31 with over 600,000,000 digits, was the smallest Fermat number whose character (prime or composite) was unknown. On April 12, 2001, Alexander Kruppa discovered that 46931635677864055013377 = 5463561471303 * 2^33 + 1 is a factor of F31. Now the smallest Fermat number of unknown character is F33, which has 2,585,827,974 digits!

Many Fermat numbers larger than F33 are known to be composite, but only because someone found a prime number that happened to divide evenly into one of these numbers. The current status of the factoring of Fermat numbers is available here.

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