Jim's Prime Page

My Page about Prime Numbers

This page uses _subscripts and ^superscripts. If you don’t see _subscripts and ^superscipts in this sentence, you may have difficulty reading this page.

The Great Internet Mersenne Prime Search

GIMPS, the “Great Internet Mersenne Prime Search”, has found thirteen Mersenne primes. See their history page for links to announcements for each of the following primes.
2^42643801 - 1 (12,837,064 digits) on April 12, 2009. For more details, see the GIMPS announcement.
2^37156667 - 1 (11,185,272 digits) on September 6, 2008. For more details, see the GIMPS announcement.
2^43112609 - 1 (12,978,189 digits) on August 23, 2008. This was the first known prime number with more than 10,000,000 digits. For more details, see the GIMPS announcement.
2^32582657 - 1 (9,808,358 digits) on September 6, 2006.
2^30402457 - 1 (9,152,052 digits) on December 15, 2005.
2^25964951 - 1 (7,816,230 digits) on February 18, 2005.
2^24036583 - 1 (7,235,733 digits) on May 15, 2004.
2^20996011 - 1 (6,320,430 digits) on November 17, 2003.
2^13466917 - 1 (4,053,946 digits) on November 14, 2001.
2^6972593 - 1 (2,098,960 digits) on June 1, 1999. This was the first known prime number with more than 1,000,000 digits.
2^3021377 - 1 (909,526 digits) in January, 1998.
2^2976221 - 1 (895,932 digits) in August, 1997.
2^1398269 - 1 (420,921 digits) in November, 1996.
I have participated in GIMPS since about September of 1996. For more information on GIMPS, including how you can join the search, see the GIMPS web site.

More about Prime Numbers

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 47 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 2^16777216 + 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.

E-mail me at jrhowell@ix.netcom.com

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This page was last updated on February 26, 2011.