# 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 = 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,

Mn = 2n − 1.

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.

## Properties of Mersenne numbers

Mersenne numbers share several properties:

Mn is a sum of binomial coefficients: ${\displaystyle M_{n}=\sum _{i=0}^{n}{n \choose i}-1}$ .

If a is a divisor of Mq (q prime) then a has the following properties: :${\displaystyle a\equiv 1{\pmod {2q}}}$ and: ${\displaystyle a\equiv \pm 1{\pmod {8}}}$ .

A theorem from Euler about numbers of the form 1+6k shows that Mq (q prime) is a prime if and only if there exists only one pair ${\displaystyle (x,y)}$ such that: ${\displaystyle M_{q}={(2x)}^{2}+3{(3y)}^{2}}$ with ${\displaystyle q\geq 5}$ . More recently, Bas Jansen has studied ${\displaystyle M_{q}=x^{2}+dy^{2}}$ for d = 0 ... 48 and has provided a new (and clearer) proof for case d = 3.

Let ${\displaystyle q=3\ {\pmod {4}}}$ be a prime. ${\displaystyle 2q+1}$ is also a prime if and only if ${\displaystyle 2q+1}$ divides ${\displaystyle M_{q}}$ .

Reix has recently found that prime and composite Mersenne numbers Mq (q prime > 3) can be written as: ${\displaystyle M_{q}={(8x)}^{2}-{(3qy)}^{2}={(1+Sq)}^{2}-{(Dq)}^{2}}$ . Obviously, if there exists only one pair (x, y), then Mq is prime.

Ramanujan has showed that the equation: ${\displaystyle M_{q}=6+x^{2}}$ has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite).

## Searching for Mersenne primes

The calculation

${\displaystyle (2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\dots +2^{(b-1)a}\right)=2^{ab}-1}$

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, ${\displaystyle 2^{11}-1=23\cdot 89}$.

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 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 M127, found by Lucas in 1876, then M61 by Pervushin in 1883. Two more - M89 and M107 - were found early in the 20th century, by 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 Lucas in 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test. Specifically, it can be shown that ${\displaystyle M_{n}=2^{n}-1}$ is prime if and only if Mn evenly divides Sn-2, where ${\displaystyle S_{0}=4}$ and for ${\displaystyle k>0}$, ${\displaystyle S_{k}=S_{k-1}^{2}-2}$.

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 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, 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, and M44497 is the first Gigantic.

As of August 2005, only 42 Mersenne primes were known; the largest known prime number (225,964,951 − 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 (GIMPS).

### List of Mersenne primes

The table below lists all known Mersenne primes (sequence A000668 in OEIS):

# n Mn Digits in Mn Date of discovery Discoverer
1 2 3 1 ancient ancient
2 3 7 1 ancient ancient
3 5 31 2 ancient ancient
4 7 127 3 ancient ancient
5 13 8191 4 1456 anonymous
6 17 131071 6 1588 Cataldi
7 19 524287 6 1588 Cataldi
8 31 2147483647 10 1772 Euler
9 61 2305843009213693951 19 1883 Pervushin
10 89 618970019…449562111 27 1911 Powers
11 107 162259276…010288127 33 1914 Powers
12 127 170141183…884105727 39 1876 Lucas
13 521 686479766…115057151 157 January 30 1952 Robinson
14 607 531137992…031728127 183 January 30 1952 Robinson
15 1,279 104079321…168729087 386 June 25 1952 Robinson
16 2,203 147597991…697771007 664 October 7 1952 Robinson
17 2,281 446087557…132836351 687 October 9 1952 Robinson
18 3,217 259117086…909315071 969 September 8 1957 Riesel
19 4,253 190797007…350484991 1,281 November 3 1961 Hurwitz
20 4,423 285542542…608580607 1,332 November 3 1961 Hurwitz
21 9,689 478220278…225754111 2,917 May 11 1963 Gillies
22 9,941 346088282…789463551 2,993 May 16 1963 Gillies
23 11,213 281411201…696392191 3,376 June 2 1963 Gillies
24 19,937 431542479…968041471 6,002 March 4 1971 Tuckerman
25 21,701 448679166…511882751 6,533 October 30 1978 Noll & Nickel
26 23,209 402874115…779264511 6,987 February 9 1979 Noll
27 44,497 854509824…011228671 13,395 April 8 1979 Nelson & Slowinski
28 86,243 536927995…433438207 25,962 September 25 1982 Slowinski
29 110,503 521928313…465515007 33,265 January 28 1988 Colquitt & Welsh
30 132,049 512740276…730061311 39,751 September 20 1983 Slowinski
31 216,091 746093103…815528447 65,050 September 6 1985 Slowinski
32 756,839 174135906…544677887 227,832 February 19 1992 Slowinski & Gage
33 859,433 129498125…500142591 258,716 January 10 1994 Slowinski & Gage
34 1,257,787 412245773…089366527 378,632 September 3 1996 Slowinski & Gage
35 1,398,269 814717564…451315711 420,921 November 13 1996 GIMPS / Joel Armengaud
36 2,976,221 623340076…729201151 895,932 August 24 1997 GIMPS / Gordon Spence
37 3,021,377 127411683…024694271 909,526 January 27 1998 GIMPS / Roland Clarkson
38 6,972,593 437075744…924193791 2,098,960 June 1 1999 GIMPS / Nayan Hajratwala
39* 13,466,917 924947738…256259071 4,053,946 November 14 2001 GIMPS / Michael Cameron
40* 20,996,011 125976895…855682047 6,320,430 November 17 2003 GIMPS / Michael Shafer
41* 24,036,583 299410429…733969407 7,235,733 May 15 2004 GIMPS / Josh Findley
42* 25,964,951 122164630…577077247 7,816,230 February 18 2005 GIMPS / Martin Nowak

*It is not known whether any undiscovered Mersenne primes exist between the 38th (M6972593) and the 42nd (M25964951) on this chart; the ranking is therefore provisional.