## 42nd Mersenne Prime (Probably) Discovered

### By Eric W. Weisstein

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 Mn = 2n - 1, giving the first few as 1, 3, 7, 15, .... Interestingly, [by] the definition of these numbers 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. [** But notice that 211-1=(23)(89) is not 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 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 M20,996,011 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.

References

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.

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