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# Mersenne prime

In mathematics, a **Mersenne prime** is a prime number that is one less than a power of two. For example, 3 = 4 − 1 = 2^{2} − 1 is a Mersenne prime; so is 7 = 8 − 1 = 2^{3} − 1. On the other hand, 15 = 16 − 1 = 2^{4} − 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 *M _{n}* can be prime only if

*n*itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true;

*M*may be composite even though

_{n}*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 *M*_{2}, *M*_{3}, *M*_{5}, *M*_{7} were known in antiquity.
The fifth, *M*_{13}, was discovered anonymously before 1461; the next two (*M*_{17} and *M*_{19}) were found by Pietro Cataldi in 1588. After more than a century *M*_{31} was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was *M*_{127}, found by Lucas in 1876, then *M*_{61} by Ivan Pervushin in 1883. Two more - *M*_{89} and *M*_{107} - 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 *M*_{67} and *M*_{257}, and omitted *M*_{61}, *M*_{89} and *M*_{107}.

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, 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 *M _{p}* evenly divides

*S*, 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].

_{p-2}The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. The first successful identification of a Mersenne prime, *M*_{521}, 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. Raphael M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, *M*_{607}, was found by the computer a little less than two hours later. Three more — *M*_{1279}, *M*_{2203}, *M*_{2281} — were found by the same program in the next several months. *M*_{4253} is the first Mersenne prime that is Titanic, *M*_{44497} is the first Gigantic and *M*_{6972593} is the first Megaprime.

The greatest Mersenne Prime so far is
[math]\displaystyle{ 2^{82\,589\,933}-1 }[/math] (M51). 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

- Great Internet Mersenne Prime Serarch (GIMPS) - home page of mersenne.org
- prime Mersenne Numbers - History, Theorems and Lists - Explanation
- Mersenne numbers - Wolfram Research/Mathematica
- prime Mersenne numbers - Wolfram Research/Mathematica
- Mersenne Prime Bibliography with Hyperlinks to original publications
- Wikipedia article on Mersenne primes (source)