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Mersenne prime

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In mathematics, a Mersenne prime is a prime 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. On the other hand, 15 = 16 − 1 = 24-1, for example, is not a prime, because 15 is divisible by 3 and 5.

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence,

[math]\displaystyle{ M_n=2^n{-}1 }[/math] .

Mersenne primes have a close 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 exists.

It is currently unknown whether there is an infinite number of Mersenne primes.

Searching for Mersenne primes

The calculation

[math]\displaystyle{ (2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+ ... +2^{(b-1)a}\right)=2^{ab}-1 }[/math]

shows that Mn can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; Mn may be composite even though n is prime. For example, [math]\displaystyle{ 2^{11} - 1 = 23 \cdot 89 }[/math].

Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.

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 Pietro Cataldi in 1588. After more than a century M31 was verified to be prime by Leonhard Euler in 1750. The next (in historical, not numerical order) was M127, found by Édouard Lucas in 1876, then M61 by Ivan Pervushin in 1883. Two more - M89 and M107 - were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.

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, though, as he mistakenly included M67 and M257, and omitted M61, M89 and M107.

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 Édouard Lucas in 1878 and improved by Derrick Henry Lehmer in the 1930s, now known as the Lucas-Lehmer test. Specifically, it can be shown that if p is an odd prime, then [math]\displaystyle{ M_p=2^p-1 }[/math] is prime if and only if Mp evenly divides [math]\displaystyle{ S_{p-2} }[/math], where [math]\displaystyle{ S_0=4 }[/math] and for [math]\displaystyle{ k\gt 0 }[/math], [math]\displaystyle{ S_k=S_{k-1}^2-2 }[/math].

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. The first successful identification of a Mersenne prime, M521, by this means was achieved at 10:00 P.M. on 1952-01-30 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 Derrick Henry Lehmer, with a computer search program written and run by Raphael M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M607, was found by the computer a little less than two hours later. Three more — M1279, M2203, M2281 — were found by the same program in the next several months. M4253 is the first Mersenne prime that is Titanic, M44497 is the first Gigantic and M6,972,593 is the first Megaprime.

The greatest Mersenne prime so far is 282,589,933-1. Like several previous Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS).

See also

External links

Number classes
General numbers
Special numbers
Prime numbers