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Mersenne prime

A Mersenne prime is a prime number that is one less than a power of two. For example, 3 = 4-1 = 2²-1 is a Mersenne prime. So is 7 = 8-1 = 2³-1. But 15 = 16-1 = 2⁴-1 isn't a prime. Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.

Mersenne primes have a deep connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection. In the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist.

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than an odd power of two. The notation M_n = 2ⁿ-1 is used. The calculation

(2^a - 1) · (1 + 2^a + 2^2a + 2^3a + ... + 2^(b-1)a) = 2^ab - 1

shows that M_n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse isn't true; M_n may be composite even though n is prime. For example, 2¹¹-1 = 23×89.

The first four Mersenne primes M₂, M₃, M₅, M₇ were known in antiquity. The fifth, M₁₃, was discovered anonymously before 1461. The next two (M₁₇ and M₁₉) were found by Cataldi in 1588. After more than a century M₃₁ was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was found by Lucas in 1876, then another by Pervushin in 1883. Two more were found early in the 20th century, by Powers in 1911 and Fauquembergue in 1914.

The numbers are named after 17th century French mathematician Marin Mersenne, who provided a list of Mersenne primes with exponents up to 257. Unfortunately his list was not correct. He mistakenly included M₆₇ and M₂₅₇, and omitted M₆₁, M₈₉ and M₁₀₉.

The best method presently known for testing the primality of Mersenne numbers is based on the computation of a recurring sequence, as developed originally by Lucas in 1878 and improved by Lehmer[?] in the 1930s. Specifically, it can be shown that M_n = 2ⁿ-1 is prime if and only if M_n evenly divides S_n-2, where S₀ = 4 and for k > 0, S_k = S_k-1²-2.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. The first successful identification of a Mersenne prime by this means was achieved at 10:00 P.M. on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis[?] at the University of California, Los Angeles, under the direction of Lehmer[?], with a computer search program written and run by Prof. R.M. Robinson[?]. It was the first Mersenne prime to be identified in thirty-eight years. The next one was found by the computer a little less than two hours later. Three more were found by the same program in the next several months.

As of December 2001, only 39 Mersenne primes were known; the largest known prime number (2^13466917-1) is a Mersenne prime. Like several previous Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search.

See also:

Fermat primes, Erdös-Borwein constant

External links

Mersenne prime section of the Prime Pages: http://www.utm.edu/research/primes/mersenne.shtml
Mersenne Prime Search home page: http://www.mersenne.org/prime.htm

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