MathWorld Headline News

44th Mersenne Prime (Probably) Discovered

By Eric W. Weisstein

September 4, 2006--Less than a year after the 43rd Mersenne prime was reported (MathWorld headline news: December 25, 2005), Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a Sep. 4 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 44th known Mersenne prime. A verification run on the number has been started, and more details will be made available when confirmation of the discovery has been completed.

[Addendum: As of September 11, the new Mersenne prime has been verified. See the MathWorld headline news, September 11, 2006.]

Mersenne numbers are numbers of the form M_n = 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 9,152,052 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 ten 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 13,476,000, while all exponents below 17,546,000 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.