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45th and 46th Mersenne Primes Found

By Eric W. Weisstein

September 16, 2008--Two years after the 44th Mersenne prime was reported (MathWorld headline news: September 11, 2006), the Great Internet Mersenne Prime Search (GIMPS) project has discovered the 45th and 46th known Mersenne primes. The discoveries were made by Edson Smith on August 23, 2008 (for the larger prime) and Hans-Michael Elvenich on September 6, 2008 (for the smaller prime), and announced by GIMPS organizer George Woltman on September 16. As with the previous Mersenne prime discovery (in which Dr. Curtis Cooper and Dr. Steven Boone who at staggering odds, were also the discoverers of the 43rd known Mersenne prime), thus proving that lightning not only strikes twice, it can double-strike tiwce! Additional details can be found in the Mersenne.org press release.

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. Mersenne primes are Mersenne numbers that are also prime, i.e., have no factors other than 1 and themselves. So, since the number 127 is prime and is a Mersenne number, it is a Mersenne prime.

The new Mersenne primes are 237,156,667 - 1 = 20225440689097733553...21340265022308220927 and 243,112,609 - 1 = 31647026933025592314...80022181166697152511 (where the ellipsis indicates that several million intervening digits have been omitted for conciseness) and have a whopping total of 11,185,272 and 12,978,189 decimal digits, respectively. Both primes therefore are not only the largest known Mersenne primes, but also the largest known primes of any kind. In fact, there is a particularly efficient and, more importantly, deterministic primality test for Mersenne numbers known as the Lucas-Lehmer test. The efficiency of this test combined with the high historical profile of the Mersenne numbers thus accounts for the fact that the eight largest known primes are all Mersenne primes (prime database).

For those curious to see the new primes in their full 11,185,272 and 12,978,189 digits of glory, the results of a short Mathematica calculation generating their decimal digits are available by downloading the notebooks mersenne45.nb and mersenne46.nb. If you do not own Mathematica, you can download a free player version to view this file. A poster featuring all 12.9 million digits of the new prime (displayed in an extremely small point size) created by Richard Crandall, discoverer of the advanced transform algorithm used by the GIMPS program, is (or will shortly be) available from Perfectly Scientific.

The twelve largest known Mersenne primes (including the latest) 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 17,001,247, while all exponents below 21,842,101 have been tested at least once.

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). However, note that the region after the 40th known Mersenne primes has not been completely searched, so while the 41st number listed is the 41st Mersenne prime discovered to date, it is not yet known if M24,036,583 is actually the 41st 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
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? 32582657 9808358 Sep. 4, 2006 Curtis Cooper and Steven Boone/GIMPS
45? 37156667 11185272 Sep. 6, 2008 Hans-Michael Elvenich/GIMPS
46? 43112609 12978189 Aug. 23, 2008 Edson Smith/GIMPS

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

Mersenne.org. "MTitanic Primes Raced to Win $100,000 Research Award." Sep. 15, 2008. http://mersenne.org/m45and46.htm