44th Mersenne Prime (Probably) Discovered

By Eric W. Weisstein

September 4, 2006--Less than a year after the 43rd Mersenne prime was reported (MathWorld headline news: December 25, 2005), Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a Sep. 4 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 44th known Mersenne prime. A verification run on the number has been started, and more details will be made available when confirmation of the discovery has been completed.

[Addendum: As of September 11, the new Mersenne prime has been verified. See the MathWorld headline news, September 11, 2006.]

Mersenne numbers are numbers of the form M n = 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, M 7 = 27 - 1 = 127 = 1111111 2 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 9,152,052 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 M 20,996,011 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 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 40? 20996011 6320430 Nov. 17, 2003 Michael Shafer/GIMPS 41? 24036583 7235733 May 15, 2004 Josh Findley/GIMPS 42? 25964951 7816230 Feb. 18, 2005 Martin Nowak/GIMPS 43? 30402457 9152052 Dec. 15, 2005 Curtis Cooper and Steven Boone/GIMPS 44? ? ? Sep. 4, 2006 GIMPS

The ten 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 13,476,000, while all exponents below 17,546,000 have been tested at least once. The candidate prime 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.