MathWorld Headline News

47th Known Mersenne Prime Apparently Discovered

By Eric W. Weisstein

June 7, 2009--Less than a year after the 45th and 46th known Mersenne primes were discovered, Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a June 7 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 47th Mersenne prime discovered. A verification run on the number has been started, and more details will be made available when confirmation of the discovery has been completed. The prime was apparently discovered in April, but was not noticed due to a configuration problem with the server that prevented a notification email being sent to the search organizers.

[Postscript: The prime has now been officially verified and announced to be M_42643801, which has 12837064 decimal digits, making it the 46th known Mersenne prime ranked by size, and hence only the second largest. It was found by Norwegian GIMPS participant Odd Magnar Strindmo.]

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, 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 12,978,189 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.

The following table summarizes all known Mersenne primes.

#		digits	year	discoverer (reference)	value
1	2	1	antiquity		3
2	3	1	antiquity		7
3	5	2	antiquity		31
4	7	3	antiquity		127
5	13	4	1461	Reguis (1536), Cataldi (1603)	8191
6	17	6	1588	Cataldi (1603)	131071
7	19	6	1588	Cataldi (1603)	524287
8	31	10	1750	Euler (1772)	2147483647
9	61	19	1883	Pervouchine (1883), Seelhoff (1886)	2305843009213693951
10	89	27	1911	Powers (1911)	618970019642690137449562111
11	107	33	1913	Powers (1914)	162259276829213363391578010288127
12	127	39	1876	Lucas (1876)	170141183460469231731687303715884105727
13	521	157	Jan. 30, 1952	Robinson (1954)	68647976601306097149...12574028291115057151
14	607	183	Jan. 30, 1952	Robinson (1954)	53113799281676709868...70835393219031728127
15	1279	386	Jun. 25, 1952	Robinson (1954)	10407932194664399081...20710555703168729087
16	2203	664	Oct. 7, 1952	Robinson (1954)	14759799152141802350...50419497686697771007
17	2281	687	Oct. 9, 1952	Robinson (1954)	44608755718375842957...64133172418132836351
18	3217	969	Sep. 8, 1957	Riesel	25911708601320262777...46160677362909315071
19	4253	1281	Nov. 3, 1961	Hurwitz	19079700752443907380...76034687815350484991
20	4423	1332	Nov. 3, 1961	Hurwitz	28554254222827961390...10231057902608580607
21	9689	2917	May 11, 1963	Gillies (1964)	47822027880546120295...18992696826225754111
22	9941	2993	May 16, 1963	Gillies (1964)	34608828249085121524...19426224883789463551
23	11213	3376	Jun. 2, 1963	Gillies (1964)	28141120136973731333...67391476087696392191
24	19937	6002	Mar. 4, 1971	Tuckerman (1971)	43154247973881626480...36741539030968041471
25	21701	6533	Oct. 30, 1978	Noll and Nickel (1980)	44867916611904333479...57410828353511882751
26	23209	6987	Feb. 9, 1979	Noll (Noll and Nickel 1980)	40287411577898877818...36743355523779264511
27	44497	13395	Apr. 8, 1979	Nelson and Slowinski	85450982430363380319...44867686961011228671
28	86243	25962	Sep. 25, 1982	Slowinski	53692799550275632152...99857021709433438207
29	110503	33265	Jan. 28, 1988	Colquitt and Welsh (1991)	52192831334175505976...69951621083465515007
30	132049	39751	Sep. 20, 1983	Slowinski	51274027626932072381...52138578455730061311
31	216091	65050	Sep. 6, 1985	Slowinski	74609310306466134368...91336204103815528447
32	756839	227832	Feb. 19, 1992	Slowinski and Gage	17413590682008709732...02603793328544677887
33	859433	258716	Jan. 10, 1994	Slowinski and Gage	12949812560420764966...02414267243500142591
34	1257787	378632	Sep. 3, 1996	Slowinski and Gage	41224577362142867472...31257188976089366527
35	1398269	420921	Nov. 12, 1996	Joel Armengaud/GIMPS	81471756441257307514...85532025868451315711
36	2976221	895832	Aug. 24, 1997	Gordon Spence/GIMPS	62334007624857864988...76506256743729201151
37	3021377	909526	Jan. 27, 1998	Roland Clarkson/GIMPS	12741168303009336743...25422631973024694271
38	6972593	2098960	Jun. 1, 1999	Nayan Hajratwala/GIMPS	43707574412708137883...35366526142924193791
39	13466917	4053946	Nov. 14, 2001	Michael Cameron/GIMPS	92494773800670132224...30073855470256259071
40	20996011	6320430	Nov. 17, 2003	Michael Shafer/GIMPS	12597689545033010502...94714065762855682047
41?	24036583	7235733	May 15, 2004	Josh Findley/GIMPS	29941042940415717208...67436921882733969407
42?	25964951	7816230	Feb. 18, 2005	Martin Nowak/GIMPS	12216463006127794810...98933257280577077247
43?	30402457	9152052	Dec. 15, 2005	Curtis Cooper and Steven Boone/GIMPS	31541647561884608093...11134297411652943871
44?	32582657	9808358	Sep. 4, 2006	Curtis Cooper and Steven Boone/GIMPS	12457502601536945540...11752880154053967871
45?	37156667	11185272	Sep. 6, 2008	Hans-Michael Elvenich/GIMPS	20225440689097733553...21340265022308220927
46?	42643801	12837064	Jun. 12, 2009	Odd Magnar Strindmo/GIMPS	16987351645274162247...84101954765562314751
47?	43112609	12978189	Aug. 23, 2008	Edson Smith/GIMPS	31647026933025592314...80022181166697152511

The 13 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 18,000,949, while all exponents below 26,181,803 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.