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

 M_p=2^p-1.\,

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

A Mersenne prime is a Mersenne number that is prime. It is known that if 2p − 1 is prime then p is prime so it makes no difference which Mersenne number definition is used. As of October 2009, only 47 Mersenne primes are known; the largest known prime number (243,112,609 − 1) is a Mersenne prime.[1] In modern times, the largest known prime has almost always been a Mersenne prime.[2] 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 base-10 digits.

Contents

About Mersenne primes

Unsolved problems in mathematics
Are there infinitely many Mersenne primes? Question mark2.svg

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite. 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 Mp to be a Mersenne prime, the exponent p 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, often 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 primality test is an efficient primality test that greatly aids this task. The 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.

Searching for Mersenne primes

The identity

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

shows that Mp can be prime only if p itself is prime—that is, the primality of p is necessary but not sufficient for Mp to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that Mp is necessarily prime if p 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 2009 are Mersenne primes.

The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Euler in 1772. 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 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[3][4] and improved by Lehmer in the 1930s, now known as the Lucas–Lehmer primality test. Specifically, it can be shown that (for p > 2) Mp = 2p − 1 is prime if and only if Mp divides Sp−2, where S0 = 4 and, for k > 0,

S_k = S_{k-1}^2-2.\
Graph of number of digits in largest known Mersenne prime by year - electronic era. Note that the vertical scale is logarithmic.

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.[5] But 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.[6] All three were the first known prime of any kind of that size.

In September 2008, mathematicians at UCLA participating in GIMPS 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, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This is the eighth Mersenne prime discovered at UCLA.[7]

On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. This report was apparently overlooked until June 4, 2009. The find was verified on June 12, 2009. The prime is 242,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is less than the largest known which was the 45th to be discovered.

Theorems about Mersenne numbers

  1. If 2p − 1 is prime, then p is prime.
    • Proof: suppose that p is composite, hence can be written p=a\cdot b with a and b > 1. As stated above, 2^{ab}-1=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right). (To check this formula, just compute the right-hand product: most terms will cancel out.) We have thus written 2ab − 1 as a product of integers > 1, Q.E.D.
  2. 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.
    • Examples: Example I: 25 − 1 = 31 is prime, and 31 is 1 plus a multiple of 2×5. Example II: 211 − 1 = 23×89, where 23 = 1 + 2×11, and 89 = 1 + 8×11.
    • Proof: If q divides 2p − 1 then 2p ≡ 1 (mod q). By Fermat's Little Theorem, 2(q − 1) ≡ 1 (mod q). Assume p and q − 1 are 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)xkp ≡ 1 (mod q). But by definition, (q − 1)xkp = 1, implying that 21 ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p and q − 1 are relatively prime is untenable. Since p is prime q − 1 must be a multiple of p.
    • Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2p − 1 must be larger than p.
  3. If p is an odd prime, then any prime q that divides 2p − 1 must be congruent to \pm 1 \pmod 8.
    • Proof: 2p + 1 = 2(mod q), so 2(p + 1) / 2 is a square root of 2 modulo q. By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to \pm 1 \pmod 8.
  4. A Mersenne prime cannot be a Wieferich prime.
    • Proof: We show if p = 2m − 1 is a Mersenne prime, then the congruence 2^{p-1} \equiv 1 \pmod {p^2} does not satisfy. By Fermat's Little theorem, m | p − 1. Now write, p − 1 = mλ. If the given congruence satisfies, then p2 | 2mλ − 1,therefore 0 \equiv (2^{m\lambda}-1)/(2^m-1)=1+2^{m}+2^{2m}+...+2^{(\lambda-1)m} \equiv -\lambda \pmod {2^m-1}. Hence 2m − 1 | λ,and therefore \lambda \geq 2^m-1. This leads to p-1 \geq m(2^m-1), which is impossible since m \geq 2.

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 little indication how he came up with his list,[8] and its rigorous verification was completed more than two centuries later.

List of known Mersenne primes

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

# p Mp Digits in Mp Date of discovery Discoverer
1 2 3 1 5th century BC[9] Ancient Greek mathematicians
2 3 7 1 5th century BC[9] Ancient Greek mathematicians
3 5 31 2 3rd century BC[9] Ancient Greek mathematicians
4 7 127 3 3rd century BC[9] Ancient Greek mathematicians
5 13 8191 4 1456 anonymous [10]
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[11]
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[9] Slowinski
31 216,091 746093103…815528447 65,050 September 1, 1985[9] Slowinski
32 756,839 174135906…544677887 227,832 February 19, 1992 Slowinski & Gage on Harwell Lab Cray-2[12]
33 859,433 129498125…500142591 258,716 January 4, 1994[13] Slowinski & Gage
34 1,257,787 412245773…089366527 378,632 September 3, 1996 Slowinski & Gage[14]
35 1,398,269 814717564…451315711 420,921 November 13, 1996 GIMPS / Joel Armengaud[15]
36 2,976,221 623340076…729201151 895,932 August 24, 1997 GIMPS / Gordon Spence[16]
37 3,021,377 127411683…024694271 909,526 January 27, 1998 GIMPS / Roland Clarkson[17]
38 6,972,593 437075744…924193791 2,098,960 June 1, 1999 GIMPS / Nayan Hajratwala[18]
39 13,466,917 924947738…256259071 4,053,946 November 14, 2001 GIMPS / Michael Cameron[19]
40[*] 20,996,011 125976895…855682047 6,320,430 November 17, 2003 GIMPS / Michael Shafer[20]
41[*] 24,036,583 299410429…733969407 7,235,733 May 15, 2004 GIMPS / Josh Findley[21]
42[*] 25,964,951 122164630…577077247 7,816,230 February 18, 2005 GIMPS / Martin Nowak[22]
43[*] 30,402,457 315416475…652943871 9,152,052 December 15, 2005 GIMPS / Curtis Cooper & Steven Boone[23]
44[*] 32,582,657 124575026…053967871 9,808,358 September 4, 2006 GIMPS / Curtis Cooper & Steven Boone[24]
45[*] 37,156,667 202254406…308220927 11,185,272 September 6, 2008 GIMPS / Hans-Michael Elvenich[25]
46[*] 42,643,801 169873516…562314751 12,837,064 April 12, 2009[**] GIMPS / Odd M. Strindmo
47[*] 43,112,609 316470269…697152511 12,978,189 August 23, 2008 GIMPS / Edson Smith[25]

 * It is not known whether any undiscovered Mersenne primes exist between the 39th (M13,466,917) and the 47th (M43,112,609) on this chart; the ranking is therefore provisional. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, the current record holder was followed by 2 smaller Mersenne primes, first 2 weeks later and then 8 months later.

 ** M42,643,801 was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad.

To help visualize the size of the 47th 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.[9]

Factorization of Mersenne numbers

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, 21039 − 1 is the record-holder,[26] 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 2010, the composite Mersenne number with largest proven prime factors is 220887 − 1, which is known to have a factor p with 6229 digits that was proven prime with ECPP.[27] The largest with probable prime factors allowed is 2684127 − 1 = 23765203727 × q, where q is a probable prime.[28]

Perfect numbers

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

2^{p-1}\cdot(2^p - 1) = M_p(M_p + 1)/2\

is an even perfect number (which is also the Mpth triangular 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, but it appears unlikely that there is one.

Generalization

The binary representation of 2p − 1 is the digit 1 repeated p times, for example, 25 − 1 = 111112 in the binary notation. A Mersenne number is therefore a repunit in base 2, and Mersenne primes are the base 2 repunit primes.

The base 2 representation of a Mersenne number shows the factorization pattern for composite exponent. For example:

 M_{35}=2^{35}-1=(11111111111111111111111111111111111)_2\,
=(11111)_2 \cdot (1000010000100001000010000100001)_2=M_5 \cdot (1000010000100001000010000100001)_2\,
=(1111111)_2 \cdot (10000001000000100000010000001)_2=M_7 \cdot (10000001000000100000010000001)_2\,
=(11111)_2 \cdot (1111111)_2 \cdot [(1000010100101010010100001)_2 - (0100001010010100101000010)_2]\,
=M_5 \cdot M_7 \cdot (100001010010101101011111)_2.\,

Mersenne numbers in nature and elsewhere

In computer science, unsigned p-bit integers can be used to express numbers up to Mp.

In the mathematical problem Tower of Hanoi, solving a puzzle with a p-disc tower requires at least Mp steps.

The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is the fifth Mersenne prime.[29] The asteroids with the previous four numbers corresponding to Mersenne primes (3 Juno, 7 Iris, 31 Euphrosyne, 127 Johanna) were already named after others.

See also

References

  1. ^ 12-million-digit prime number sets record, nets $100,000 prize
  2. ^ The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "The Largest Known Prime by Year: A Brief History" from the Prime Pages website, University of Tennessee at Martin.
  3. ^ The Prime Pages, The Largest Known Prime by Year: A Brief History.
  4. ^ Prime Curios!, 17014...05727 (39-digits).
  5. ^ Brian Napper, The Mathematics Department and the Mark 1.
  6. ^ The Prime Pages, The Prime Glossary: megaprime.
  7. ^ UCLA mathematicians discover a 13-million-digit prime number, Los Angeles Times, September 27, 2008
  8. ^ The Prime Pages, Mersenne's conjecture.
  9. ^ a b c d e f g Landon Curt Noll, Mersenne Prime Digits and Names.
  10. ^ The Prime Pages, Mersenne Primes: History, Theorems and Lists.
  11. ^ The Prime Pages, M107: Fauquembergue or Powers?.
  12. ^ The Prime Pages, The finding of the 32nd Mersenne.
  13. ^ Chris Caldwell, The Largest Known Primes.
  14. ^ The Prime Pages, A Prime of Record Size! 21257787-1.
  15. ^ GIMPS Discovers 35th Mersenne Prime.
  16. ^ GIMPS Discovers 36th Known Mersenne Prime.
  17. ^ GIMPS Discovers 37th Known Mersenne Prime.
  18. ^ GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
  19. ^ GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
  20. ^ GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
  21. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583-1.
  22. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951-1.
  23. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457-1.
  24. ^ GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657-1.
  25. ^ a b Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
  26. ^ Paul Zimmermann, "Integer Factoring Records".
  27. ^ Chris Caldwell, The Top Twenty: Mersenne cofactor at The Prime Pages.
  28. ^ Donovan Johnson, "Largest known probable prime Mersenne Cofactors".
  29. ^ JPL Small-Body Database Browser

External links

MathWorld links

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