February 18, 2005--Less than a year after the 41st Mersenne prime was reported
(*MathWorld* headline news: June
1, 2004), Great Internet Mersenne Prime Search (GIMPS) project
organizer George Woltman is reporting in a Feb. 18 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 42nd known
Mersenne prime, as well as the largest prime
number known of any kind.

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

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 7,235,733
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.

# | p | 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 | 1952 | Lehmer 1952-3, Robinson 1952 |

- | --- | --- | --- | [**table abbreviated by entries 14-33**] |

34 | 1257787 | 378632 | 1996 | Slowinski and Gage |

35 | 1398269 | 420921 | 1996 | Armengaud, Woltman, et al. |

36 | 2976221 | 895832 | 1997 | Spence, Woltman, GIMPS (Devlin 1997) |

37 | 3021377 | 909526 | 1998 | Clarkson, Woltman, Kurowski, GIMPS |

38 | 6972593 | 2098960 | 1999 | Hajratwala, Woltman, Kurowski, GIMPS |

39 | 13466917 | 4053946 | 2001 | Cameron, Woltman, GIMPS (Whitehouse 2001, Weisstein 2001ab) |

40? | 20996011 | 6320430 | 2003 | Shafer, GIMPS (Weisstein 2003ab) |

41? | 24036583 | 7235733 | 2004 | Findley, GIMPS (Weisstein 2004) |

42? | ? | <10000000 | 2005 | GIMPS |

The eight 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 9,889,900, while all exponents below 14,135,900 have been
tested at least once. Although the candidate prime was flagged prime by
an experienced GIMPS volunteer, it 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.

While the exact exponent of the new find has not yet been made public,
GIMPS organizer George Woltman reported that if the new candidate is
confirmed, it would be the largest known prime, which would mean it has
7,235,733 or more digits. Woltman also noted that it has fewer than
10 million digits (a holy grail for prime searchers), meaning that the
new candidate has exponent *n* somewhere between 24,036,584 and
33,219,253. Woltman is currently attempting to reproduce the find from
the user's save file, thus eliminating any chance of the report
being erroneous.

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

Weisstein, E. W. "*MathWorld* Headline News: 40th Mersenne Prime Announced."
Dec. 2, 2003.

http://mathworld.wolfram.com/news/2003-12-02/mersenne

Weisstein, E. W. "*MathWorld* Headline News: 41st Mersenne Prime Announced."
Jun. 1, 2004.

http://mathworld.wolfram.com/news/2004-06-01/mersenne

Woltman, G. "New Mersenne Prime?!" Message to The Great Internet Mersenne Prime Search List. Feb. 18, 2005.