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MathWorld Headline News


42nd Mersenne Prime (Probably) Discovered

By Eric W. Weisstein

February 18, 2005--Less than a year after the 41st Mersenne prime was reported (MathWorld headline news: June 1, 2004), Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a Feb. 18 email to the GIMPS mailing list that a new Mersenne number has been flagged as prime and reported to the project's server. If verified, this would be the 42nd known Mersenne prime, as well as the largest prime number known of any kind.

[Addendum: As of February 25, the new Mersenne prime has been verified, as communicated by Woltman in a Feb. 25 email. See MathWorld Headline News, February 26, 2005.]

Mersenne numbers are numbers of the form Mn = 2n - 1, giving the first few as 1, 3, 7, 15, 31, 63, 127, .... Interestingly, the definition of these numbers therefore means that the nth Mersenne number is simply a string of n 1s when represented in binary. For example, M7 = 27 - 1 = 127 = 11111112 is a Mersenne number. In fact, since 127 is also prime, 127 is also a Mersenne prime.

The study of such numbers has a long and interesting history, and the search for Mersenne numbers that are prime has been a computationally challenging exercise requiring the world's fastest computers. Mersenne primes are intimately connected with so-called perfect numbers, which were extensively studied by the ancient Greeks, including by Euclid. A complete list of indices n of the previously known Mersenne primes is given in the table below (as well as by sequence A000043 in Neil Sloane's On-Line Encyclopedia of Integer Sequences). The last of these has a whopping 7,235,733 decimal digits. However, note that the region between the 39th and 40th known Mersenne primes has not been completely searched, so it is not known if M20,996,011 is actually the 40th Mersenne prime.

# n digits year discoverer (reference)
1 2 1 antiquity  
2 3 1 antiquity  
3 5 2 antiquity  
4 7 3 antiquity  
5 13 4 1461 Reguis (1536), Cataldi (1603)
6 17 6 1588 Cataldi (1603)
7 19 6 1588 Cataldi (1603)
8 31 10 1750 Euler (1772)
9 61 19 1883 Pervouchine (1883), Seelhoff (1886)
10 89 27 1911 Powers (1911)
11 107 33 1913 Powers (1914)
12 127 39 1876 Lucas (1876)
13 521 157 Jan. 30, 1952 Robinson
14 607 183 Jan. 30, 1952 Robinson
15 1279 386 Jan. 30, 1952 Robinson
16 2203 664 Jan. 30, 1952 Robinson
17 2281 687 Jan. 30, 1952 Robinson
18 3217 969 Sep. 8, 1957 Riesel
19 4253 1281 Nov. 3, 1961 Hurwitz
20 4423 1332 Nov. 3, 1961 Hurwitz
21 9689 2917 May 11, 1963 Gillies (1964)
22 9941 2993 May 16, 1963 Gillies (1964)
23 11213 3376 Jun. 2, 1963 Gillies (1964)
24 19937 6002 Mar. 4, 1971 Tuckerman (1971)
25 21701 6533 Oct. 30, 1978 Noll and Nickel (1980)
26 23209 6987 Feb. 9, 1979 Noll (Noll and Nickel 1980)
27 44497 13395 Apr. 8, 1979 Nelson and Slowinski (Slowinski 1978-79)
28 86243 25962 Sep. 25, 1982 Slowinski
29 110503 33265 Jan. 28, 1988 Colquitt and Welsh (1991)
30 132049 39751 Sep. 20, 1983 Slowinski
31 216091 65050 Sep. 6, 1985 Slowinski
32 756839 227832 Feb. 19, 1992 Slowinski and Gage
33 859433 258716 Jan. 10, 1994 Slowinski and Gage
34 1257787 378632 Sep. 3, 1996 Slowinski and Gage
35 1398269 420921 Nov. 12, 1996 Joel Armengaud/GIMPS
36 2976221 895832 Aug. 24, 1997 Gordon Spence/GIMPS (Devlin 1997)
37 3021377 909526 Jan. 27, 1998 Roland Clarkson/GIMPS
38 6972593 2098960 Jun. 1, 1999 Nayan Hajratwala/GIMPS
39 13466917 4053946 Nov. 14, 2001 Michael Cameron/GIMPS (Whitehouse 2001, Weisstein 2001)
40? 20996011 6320430 Nov. 17, 2003 Michael Shafer/GIMPS (Weisstein 2003)
41? 24036583 7235733 May 15, 2004 Josh Findley/GIMPS (Weisstein 2004)
42? ? <10000000 Feb. 18, 2005 GIMPS

The eight largest known Mersenne primes (including the latest candidate) have all been discovered by GIMPS, which is a distributed computing project being undertaken by an international collaboration of volunteers. Thus far, GIMPS participants have tested and double-checked all exponents n below 9,889,900, while all exponents below 14,135,900 have been tested at least once. Although the candidate prime was flagged prime by an experienced GIMPS volunteer, it has yet to be verified by independent software running on different hardware. If confirmed, GIMPS will make an official press release that will reveal the number and the name of the lucky discoverer.

While the exact exponent of the new find has not yet been made public, GIMPS organizer George Woltman reported that if the new candidate is confirmed, it would be the largest known prime, which would mean it has 7,235,733 or more digits. Woltman also noted that it has fewer than 10 million digits (a holy grail for prime searchers), meaning that the new candidate has exponent n somewhere between 24,036,584 and 33,219,253. Woltman is currently attempting to reproduce the find from the user's save file, thus eliminating any chance of the report being erroneous.

References

Caldwell, C. K. "The Largest Known Primes." http://www.utm.edu/research/primes/largest.html

GIMPS: The Great Internet Mersenne Prime Search. http://www.mersenne.org

GIMPS: The Great Internet Mersenne Prime Search Status. http://www.mersenne.org/status.htm

Weisstein, E. W. "MathWorld Headline News: 40th Mersenne Prime Announced." Dec. 2, 2003. http://mathworld.wolfram.com/news/2003-12-02/mersenne

Weisstein, E. W. "MathWorld Headline News: 41st Mersenne Prime Announced." Jun. 1, 2004. http://mathworld.wolfram.com/news/2004-06-01/mersenne

Woltman, G. "New Mersenne Prime?!" Message to The Great Internet Mersenne Prime Search List. Feb. 18, 2005.

Woltman, G. "42nd Mersenne Prime." Message to The Great Internet Mersenne Prime Search List. Feb. 25, 2005.