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,
- M_{n} = 2^{n} − 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.
Contents
Properties of Mersenne numbers
Mersenne numbers share several properties:
M_{n} is a sum of binomial coefficients: .
If a is a divisor of M_{q} (q prime) then a has the following properties: : and: .
A theorem from Euler about numbers of the form 1+6k shows that M_{q} (q prime) is a prime if and only if there exists only one pair such that: with . More recently, Bas Jansen has studied for d = 0 ... 48 and has provided a new (and clearer) proof for case d = 3.
Let be a prime. is also a prime if and only if divides .
Reix has recently found that prime and composite Mersenne numbers M_{q} (q prime > 3) can be written as: . Obviously, if there exists only one pair (x, y), then M_{q} is prime.
Ramanujan has showed that the equation: has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite).
Searching for Mersenne primes
The calculation
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_{n} may be composite even though n is prime. For example, .
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 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 Pervushin in 1883. Two more - M_{89} and M_{107} - 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 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 is prime if and only if M_{n} evenly divides S_{n-2}, where and for , .
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. 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, and M_{44497} is the first Gigantic.
As of August 2005, only 42 Mersenne primes were known; the largest known prime number (2^{25,964,951} − 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 | M_{n} | Digits in M_{n} | 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 (M_{6972593}) and the 42nd (M_{25964951}) on this chart; the ranking is therefore provisional.
See also
- Fermat prime
- Erdős–Borwein constant
- Great Internet Mersenne Prime Search
- New Mersenne conjecture
- Prime95 / MPrime
- Lucas-Lehmer test for Mersenne primes
- Double Mersenne number
- Mersenne twister
External links
- Great Internet Mersenne Prime Serarch (GIMPS) Orlando Florida - home page of mersenne.org
- prime Mersenne Numbers - History, Theorems and Lists Explanation
- GIMPS Mersenne Prime - status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 39-42ff
- Mersenne numbers - Wolfram Research/Mathematica
- prime Mersenne numbers - Wolfram Research/Mathematica
- M_{q} = (8x)^2 - (3qy)^2 Mersenne Proof (pdf)
- M_{q} = x^2 + d.y^2 Math Thesis (ps)
- http://www.utm.edu/research/primes/mersenne/LukeMirror/biblio.htm Mersenne Prime Bibliography] with Hyperlinks to original publications
- dpa - reportage about prime mersenne number - detection in detail (German)
- Mersenne prime Wiki
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