Definitions

# Mersenne prime

In mathematics, a Mersenne number is a positive integer that is one less than a power of two:

$M_n=2^n-1.,$

Some definitions of Mersenne numbers require that the exponent n be prime.

A Mersenne prime is a Mersenne number that is prime., only 46 Mersenne primes are known; the largest known prime number (243,112,609 − 1) is a Mersenne prime, and in modern times, the largest known prime has almost always been a Mersenne prime. Like several previously-discovered Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS). It was the first known prime number with more than 10 million digits.

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is infinite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.

A basic theorem about Mersenne numbers states that in order for Mn to be a Mersenne prime, the exponent n itself must be a prime number. This rules out primality for numbers such as M4 = 24−1 = 15: since the exponent 4=2×2 is composite, the theorem predicts that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are

M2 = 3, M3 = 7, M5 = 31.

While it is true that only Mersenne numbers Mp, where p = 2, 3, 5, … could be prime, it may nevertheless turn out that Mp is not prime even for a prime exponent p. The smallest counterexample is the Mersenne number

M11 = 211 − 1 = 2047 = 23 × 89,

which is not prime, even though 11 is a prime number. The lack of an obvious rule to determine whether a given Mersenne number is prime makes the search for Mersenne primes an interesting task, which becomes difficult very quickly, since Mersenne numbers grow very rapidly. The Lucas–Lehmer test for Mersenne numbers is an efficient primality test that greatly aids this task. Search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.

Mersenne primes are used in pseudorandom number generators such as Mersenne Twister, Park–Miller RNG, Generalized Shift Register, and Fibonacci RNG.

## Searching for Mersenne primes

The identity

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

shows that Mn can be prime only if n itself is prime—that is, the primality of n is necessary but not sufficient for Mn to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that Mn is necessarily prime if n is prime, is false. The smallest counterexample is 211 − 1 = 2,047 = 23×89, a composite number.

Fast algorithms for finding Mersenne primes are available, and the largest known prime numbers as of 2008 are Mersenne primes.

The first four Mersenne primes $M_2=3$, $M_3=7$, $M_5=31$ and $M_7=127$ were known in antiquity. The fifth, $M_\left\{13\right\}=8191$, was discovered anonymously before 1461; the next two ($M_\left\{17\right\}$ and $M_\left\{19\right\}$) were found by Cataldi in 1588. After nearly two centuries, $M_\left\{31\right\}$ was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was $M_\left\{127\right\}$, found by Lucas in 1876, then $M_\left\{61\right\}$ by Pervushin in 1883. Two more ($M_\left\{89\right\}$ and $M_\left\{107\right\}$) were found early in the 20th century, by Powers in 1911 and 1914, respectively.

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 1856 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test for Mersenne numbers. Specifically, it can be shown that (for $n>2$) $M_n=2^n-1$ is prime if and only if Mn divides Sn−2, where $S_0=4$ and for $k>0$, $S_k=S_\left\{k-1\right\}^2-2$.

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, M44497 is the first gigantic, and M6,972,593 was the first megaprime to be discovered, being a prime with at least 1,000,000 digits. All three were the first known prime of any kind of that size.

In September 2008, mathematicians at UCLA participating in GIMPS appear to have won part of a \$100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23. This is the eighth Mersenne prime discovered at UCLA.

$c^n-d^n=\left(c-d\right)sum_\left\{k=0\right\}^\left\{n-1\right\} c^kd^\left\{n-1-k\right\},$

or

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

by setting c = 2a, d = 1, and n = b

proof

$\left(a-b\right)sum_\left\{k=0\right\}^\left\{n-1\right\}a^kb^\left\{n-1-k\right\}$

$=sum_\left\{k=0\right\}^\left\{n-1\right\}a^\left\{k+1\right\}b^\left\{n-1-k\right\}-sum_\left\{k=0\right\}^\left\{n-1\right\}a^kb^\left\{n-k\right\}$

$=a^n+sum_\left\{k=1\right\}^\left\{n-1\right\}a^kb^\left\{n-k\right\}-sum_\left\{k=1\right\}^\left\{n-1\right\}a^kb^\left\{n-k\right\}-b^n$

$=a^n-b^n$

• 2) If 2n − 1 is prime, then n is prime.

proof

By

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

If n is not prime, or n = ab where 1 < ab < n. Therefore, 2a − 1 would divide 2n − 1, or 2n − 1 is not prime.

• 3) If p is an odd prime, then any prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 1 is prime. Example I: 25 − 1 = 31

is prime, and 31 is 1 plus a multiple of 2×5. Example II: 211 − 1 = 23×89', 23 = 1 + 2×11, and 89 = 1 + 8×11, and also 23×89 = 1 + 186×11.

proof

If q divides 2p − 1 then 2p ≡ 1 (mod q). By Fermat's Little Theorem, 2(q − 1) ≡ 1 (mod q). Assume there exists such a p which does not divide q − 1. Then as p and q − 1 must be relatively prime, a similar application of Fermat's Little Theorem says that (q − 1)(p − 1) ≡ 1 (mod p). Thus there is a number x ≡ (q − 1)(p − 2) for which (q − 1)·x ≡ 1 (mod p), and therefore a number k for which (q − 1)·x − 1 = kp. Since 2(q − 1) ≡ 1 (mod q), raising both sides of the congruence to the power x gives 2(q − 1)x ≡ 1, and since 2p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2kp ≡ 1. Thus 2(q − 1)x ÷ 2kp = 2(q − 1)x − kp ≡ 1 (mod q). But by definition, (q − 1)x − kp = 1, implying that 21 ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p does not divide q − 1 is untenable.

• 4) If p is an odd prime, then any prime q that divides $2^p-1$ must be congruent to $pm 1 pmod 8$. Proof: $2^\left\{p+1\right\} = 2 pmod q$, so $2^\left\{\left(p+1\right)/2\right\}$ is a square root of 2 modulo $q$. By quadratic reciprocity, any prime modulo which two has a square root is congruent to $pm 1 pmod 8$.

## History

Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers. They are named after 17th century French scholar Marin Mersenne, who compiled a list of Mersenne primes with exponents up to 257. His list was only partially correct, as Mersenne mistakenly included M67 and M257 (which are composite), and omitted M61, M89, and M107 (which are prime). Mersenne gave no indication how he came up with his list, and its rigorous verification was completed more than two centuries later.

## List of known Mersenne primes

The table below lists all known Mersenne primes :
# n Mn Digits in Mn Date of discovery Discoverer
1 2 3 1 5th century BC Ancient Greek mathematicians
2 3 7 1 5th century BC Ancient Greek mathematicians
3 5 31 2 3rd century BC Ancient Greek mathematicians
4 7 127 3 3rd century BC Ancient Greek mathematicians
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 19 1983 Slowinski
31 216,091 746093103…815528447 65,050 September 1 1985 Slowinski
32 756,839 174135906…544677887 227,832 February 19 1992 Slowinski & Gage on Harwell Lab Cray-2
33 859,433 129498125…500142591 258,716 January 4 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
43 30,402,457 315416475…652943871 9,152,052 December 15 2005 GIMPS / Curtis Cooper & Steven Boone
44 32,582,657 124575026…053967871 9,808,358 September 4 2006 GIMPS / Curtis Cooper & Steven Boone
45 37,156,667 202254406…308220927 11,185,272 September 6 2008 GIMPS / Hans-Michael Elvenich
46 43,112,609 316470269…697152511 12,978,189 August 23 2008 GIMPS / Edson Smith
It is not known whether any undiscovered Mersenne primes exist between the 39th (M13,466,917) and the 46th (M43,112,609) on this chart; the ranking is therefore provisional. For a historical example, note that the 29th Mersenne prime was discovered after the 30th and the 31st. It is also remarkable that the current record holder was followed 14 days later by a smaller Mersenne prime.

To help visualize the size of the 46th known Mersenne prime, it would require 3,461 pages to display the number in base 10 with 75 digits per line and 50 lines per page.

## Factorization of Mersenne numbers

Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorised has been a Mersenne number. As of March 2007, $2^\left\{1039\right\}-1$ is the record-holder, after a calculation taking about a year on a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland. See integer factorization records for links to more information.

## Perfect numbers

Mersenne primes are interesting to many for their connection to perfect numbers. In the 4th century BC, Euclid demonstrated that if Mn is a Mersenne prime then

2n−1×(2n−1) = Mn(Mn+1)/2
is an even perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers.

## Generalization

The binary representation of 2n − 1 is the digit 1 repeated n times, for example, 25 − 1 = 111112 in the binary notation. The Mersenne primes are therefore the base-2 repunit primes.