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## 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 M42643801, 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 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 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 M20,996,011 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.

References

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

GIMPS: The Great Internet Mersenne Prime Search. "47th Known Mersenne Prime Found!" http://www.mersenne.org

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

Woltman, G. "New Mersenne Prime?" Message to The Great Internet Mersenne Prime Search List. Jun. 4, 2009. http://www.mail-archive.com/prime@hogranch.com/msg02351.html

Woltman, G. "New Mersenne Prime?" Message to The Great Internet Mersenne Prime Search List. Jun. 7, 2009. http://www.mail-archive.com/prime@hogranch.com/msg02362.html

Woltman, G. "It's Official - 47th Mersenne Prime Found" Message to The Great Internet Mersenne Prime Search List. Jun. 12, 2009. http://www.mail-archive.com/prime@hogranch.com/msg02379.html