In mathematics, a Mersenne number is a positive integer that is one less than a power of two:
Some definitions of Mersenne numbers require that the exponent n be prime.
A Mersenne prime is a Mersenne number that is prime. As of December 2008^{[ref]}, only 46 Mersenne primes are known; the largest known prime number (2^{43,112,609} − 1) is a Mersenne prime, and in modern times, the largest known prime has almost always been a Mersenne prime.^{[1]} 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.
In computer science, unsigned n-bit integers can be used to express numbers up to M_{n}.
In the mathematical problem Tower of Hanoi, solving a puzzle with an n-disc tower requires at least M_{n} steps.
Are there infinitely many Mersenne primes?
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 M_{n} to be a Mersenne prime, the exponent n itself must be a prime number. This rules out primality for numbers such as M_{4} = 2^{4}−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
While it is true that only Mersenne numbers M_{p}, where p = 2, 3, 5, … could be prime, it may nevertheless turn out that M_{p} is not prime even for a prime exponent p. The smallest counterexample is the Mersenne number
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 the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG.
The identity
shows that M_{n} can be prime only if n itself is prime—that is, the primality of n is necessary but not sufficient for M_{n} to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that M_{n} is necessarily prime if n is prime, is false. The smallest counterexample is 2^{11} − 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 , , and were known in antiquity. The fifth, , was discovered anonymously before 1461; the next two ( and ) were found by Cataldi in 1588. After nearly two centuries, was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was , found by Lucas in 1876, then by Pervushin in 1883. Two more ( and ) 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^{[2]}^{[3]} and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test for Mersenne numbers. Specifically, it can be shown that (for ) is prime if and only if M_{n} divides S_{n−2}, where and for , .
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949.^{[4]} But 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. R.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_{6,972,593} was the first megaprime to be discovered, being a prime with at least 1,000,000 digits.^{[5]} 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.^{[6]}
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 M_{67} and M_{257} (which are composite), and omitted M_{61}, M_{89}, and M_{107} (which are prime). Mersenne gave little indication how he came up with his list^{[7]}, and its rigorous verification was completed more than two centuries later.
The table below lists all known Mersenne primes (sequence A000668 in the OEIS):
# | p | M_{p} | Digits in M_{p} | Date of discovery | Discoverer |
---|---|---|---|---|---|
1 | 2 | 3 | 1 | 5th century BC^{[8]} | Ancient Greek mathematicians |
2 | 3 | 7 | 1 | 5th century BC^{[8]} | Ancient Greek mathematicians |
3 | 5 | 31 | 2 | 3rd century BC^{[8]} | Ancient Greek mathematicians |
4 | 7 | 127 | 3 | 3rd century BC^{[8]} | Ancient Greek mathematicians |
5 | 13 | 8191 | 4 | 1456 | anonymous ^{[9]} |
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^{[10]} |
12 | 127 | 170141183…884105727 | 39 | 1876 | Lucas |
13 | 521 | 686479766…115057151 | 157 | January 30, 1952 | Robinson, using SWAC (computer) |
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, using BESK |
19 | 4,253 | 190797007…350484991 | 1,281 | November 3, 1961 | Hurwitz, using IBM 7090 |
20 | 4,423 | 285542542…608580607 | 1,332 | November 3, 1961 | Hurwitz |
21 | 9,689 | 478220278…225754111 | 2,917 | May 11, 1963 | Gillies, using ILLIAC II |
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, using IBM 360/91 |
25 | 21,701 | 448679166…511882751 | 6,533 | October 30, 1978 | Noll & Nickel, using CDC Cyber 174 |
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^{[8]} | Slowinski |
31 | 216,091 | 746093103…815528447 | 65,050 | September 1, 1985^{[8]} | Slowinski |
32 | 756,839 | 174135906…544677887 | 227,832 | February 19, 1992 | Slowinski & Gage on Harwell Lab Cray-2^{[11]} |
33 | 859,433 | 129498125…500142591 | 258,716 | January 4, 1994^{[12]} | Slowinski & Gage |
34 | 1,257,787 | 412245773…089366527 | 378,632 | September 3, 1996 | Slowinski & Gage^{[13]} |
35 | 1,398,269 | 814717564…451315711 | 420,921 | November 13, 1996 | GIMPS / Joel Armengaud^{[14]} |
36 | 2,976,221 | 623340076…729201151 | 895,932 | August 24, 1997 | GIMPS / Gordon Spence^{[15]} |
37 | 3,021,377 | 127411683…024694271 | 909,526 | January 27, 1998 | GIMPS / Roland Clarkson^{[16]} |
38 | 6,972,593 | 437075744…924193791 | 2,098,960 | June 1, 1999 | GIMPS / Nayan Hajratwala^{[17]} |
39 | 13,466,917 | 924947738…256259071 | 4,053,946 | November 14, 2001 | GIMPS / Michael Cameron^{[18]} |
40^{[*]} | 20,996,011 | 125976895…855682047 | 6,320,430 | November 17, 2003 | GIMPS / Michael Shafer^{[19]} |
41^{[*]} | 24,036,583 | 299410429…733969407 | 7,235,733 | May 15, 2004 | GIMPS / Josh Findley^{[20]} |
42^{[*]} | 25,964,951 | 122164630…577077247 | 7,816,230 | February 18, 2005 | GIMPS / Martin Nowak^{[21]} |
43^{[*]} | 30,402,457 | 315416475…652943871 | 9,152,052 | December 15, 2005 | GIMPS / Curtis Cooper & Steven Boone^{[22]} |
44^{[*]} | 32,582,657 | 124575026…053967871 | 9,808,358 | September 4, 2006 | GIMPS / Curtis Cooper & Steven Boone^{[23]} |
45^{[*]} | 37,156,667 | 202254406…308220927 | 11,185,272 | September 6, 2008 | GIMPS / Hans-Michael Elvenich^{[24]} |
46^{[*]} | 43,112,609 | 316470269…697152511 | 12,978,189 | August 23, 2008 | GIMPS / Edson Smith^{[24]} |
^{ *} It is not known whether any undiscovered Mersenne primes exist between the 39th (M_{13,466,917}) and the 46th (M_{43,112,609}) 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. ^{[8]}
The factorization of a prime number is by definition the number itself. This section is about composite numbers. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of March 2007^{[update]}, 2^{1039}−1 is the record-holder,^{[25]} 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. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor. As of 2008^{[update]}, the largest composite Mersenne number with proven prime factors is 2^{17029}−1 = 418879343 × p, where p was proven prime with ECPP.^{[26]} The largest with probable prime factors allowed is 2^{173867}−1 = 52536637502689 × q, where q is a probable prime.^{[27]}
Mersenne primes are interesting to many for their connection to perfect numbers. In the 4th century BC, Euclid demonstrated that if M_{n} is a Mersenne prime then
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.
The binary representation of 2^{n} − 1 is the digit 1 repeated n times, for example, 2^{5} − 1 = 11111_{2} in the binary notation. The Mersenne primes are therefore the base-2 repunit primes.