MathWorld Headline News

43rd Mersenne Prime (Probably) Discovered

By Eric W. Weisstein

December 19, 2005--Less than a year after the 42nd Mersenne prime was reported (MathWorld headline news: February 18, 2005), Great Internet Mersenne Prime Search (GIMPS) project organizer George Woltman is reporting in a Dec. 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 43rd known Mersenne prime. A verification run on the number has been started, and will take a week or two to complete.

[Addendum: As of December 25, the new Mersenne prime has been verified. See the MathWorld headline news story for more details.]

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 7,816,230 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 nine 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 11,145,000, while all exponents below 15,464,000 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 the new candidate 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. Because Woltman conspicuously did not state that the current candidate would be the largest known prime, it is likely that the 43rd known Mersenne prime is smallerthan the 42nd. Woltman is currently attempting to reproduce the find from the user's save file, thus eliminating any chance of the report being erroneous.