*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*, 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.]

_{42643801}
Mersenne numbers are numbers of the form *M _{n}*
= 2

^{n}- 1, giving the first few as 1, 3, 7, 15, 31, 63, 127, .... Interestingly, the definition of these numbers therefore means that the

*n*th Mersenne number is simply a string of

*n*1s when represented in binary. For example,

*M*

_{7}= 2

^{7}- 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
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.

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