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

^{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
href=”http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000043″

target=”_blank”>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*

_{ style=”vertical-align:sub, font-size:10pt;”>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 |

14 | 607 | 183 | 1952 | Lehmer 1952-3, Robinson 1952 |

15 | 1279 | 386 | 1952 | Lehmer 1952-3, Robinson 1952 |

16 | 2203 | 664 | 1952 | Lehmer 1952-3, Robinson 1952 |

17 | 2281 | 687 | 1952 | Lehmer 1952-3, Robinson 1952 |

18 | 3217 | 969 | 1957 | Riesel 1957 |

19 | 4253 | 1281 | 1961 | Hurwitz 1961 |

20 | 4423 | 1332 | 1961 | Hurwitz 1961 |

21 | 9689 | 2917 | 1963 | Gillies 1964 |

22 | 9941 | 2993 | 1963 | Gillies 1964 |

23 | 11213 | 3376 | 1963 | Gillies 1964 |

24 | 19937 | 6002 | 1971 | Tuckerman 1971 |

25 | 21701 | 6533 | 1978 | Noll and Nickel 1980 |

26 | 23209 | 6987 | 1979 | Noll 1980 |

27 | 44497 | 13395 | 1979 | Nelson and Slowinski 1979 |

28 | 86243 | 25962 | 1982 | Slowinski 1982 |

29 | 110503 | 33265 | 1988 | Colquitt and Welsh 1991 |

30 | 132049 | 39751 | 1983 | Slowinski 1988 |

31 | 216091 | 65050 | 1985 | Slowinski 1989 |

32 | 756839 | 227832 | 1992 | Gage and Slowinski 1992 |

33 | 859433 | 258716 | 1994 | Gage and Slowinski 1994 |

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.

AnonymousBut will it impress the babes?