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# Mersenne prime

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Named after 17th-century French mathematician [Marin Mersenne]?, 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 is not a prime.

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 B.C. Euclid demonstrated that if M is a Mersenne prime then M*(M+1)/2 is a perfect number. Two millenia 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 Mn = 2n-1 is used. It is easily shown that Mp can only be prime if p itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; Mp may be composite even though p is prime. For example, 211-1 = 23*89.

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 Mn = 2n-1 is prime if and only if Mn evenly divides Sn-2, where S0 = 4 and for k > 0, Sk = Sk-12-2.

Mersenne did not conceive these numbers, of course, but did provide a list of exponents for Mersenne primes with exponents up to 257. Unfortunately his list was not correct. He mistakenly included 67 and 257, and omitted 61, 89 and 109.

The first four Mersenne primes were known in antiquity. The fifth (213-1) was discovered anonymously before 1461. The next two were found by Cataldi in 1588. After more than a century another was found 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 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 Standard's Western Automatic Computer (SWAC) at the Institute for Numerical Analysis on the Los Angeles campus of the University of California, under the direction of D.H. 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 March 2001, only 38 Mersenne primes were known; the largest known prime number (26972593-1) is a Mersenne prime. It was discovered by a [distributed computing]? project on the Internet, known as the Great Internet Mersenne Prime Search. See the project home page at [http://www.mersenne.org/prime.htm] for much, much more information on the properties of Mersenne primes, biographical information about Marin Mersenne, and the current status of the search for new primes.

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