MathWorld Headline News

40th Mersenne Prime (Probably) Discovered

By Eric W. Weisstein with contributions by Ed Pegg, Jr.

November 19, 2003--Almost exactly two years after the 39th largest Mersenne prime was reported (MathWorld headline news: November 14, 2001 announcement; December 5, 2001 confirmation), a notice on the Great Internet Mersenne Prime Search (GIMPS) website is reporting that a new Mersenne number passed the Lucas-Lehmer test on November 17, 2003, thus identifying it as a prime number. This would make it the 40th known Mersenne prime.

Mersenne numbers are numbers of the form M_n = 2ⁿ - 1. For example, M₇ = 2⁷ - 1 = 127 is a Mersenne number.

The study of such numbers has a long and interesting history, and the search for Mersenne numbers that are prime (so-called Mersenne primes) has been a computationally challenging exercise requiring the world's fastest computers. The complete list of indices n for previously known Mersenne primes is given by n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, and 13466917 (Sloane's A000043). The last of these has a whopping 4,053,946 digits. However, the region between the last two previously known Mersenne primes has not been completely searched, so it is not known if M_13466917 is actually the 39th Mersenne prime.

The six largest known Mersenne primes (including the latest candidate) have been discovered by an international collaboration of GIMPS volunteers. Thus far, the GIMPS participants have tested and double-checked all exponents below 7,137,900 and tested all exponents below 10,412,700 at least once. The candidate prime was flagged prime by a GIMPS volunteer on November 17, but the number has yet to be verified as prime 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. That announcement is anticipated in early December.

While the exact exponent of the new find has not yet been made public, GIMPS organizer George Woltman announced in an email message to the Mersenne Prime Mailing List that the new candidate has between five and 10 million digits, which would place the exponent somewhere between 16,609,643 and 33,219,253.

Interestingly, the fifth through 10th largest known primes have all been discovered in the year 2003, with the four largest confirmed Mersenne primes (discovered between 1997 and 2001) holding down the top four places (Caldwell). The largest previous prime found in 2003 is the generalized Fermat number 1176694¹³¹⁰⁷² + 1, discovered by Daniel Heuer on September 22, 2003 (Gallot).

An interesting listing of internet-based distributed computing projects in mathematics is maintained by Aspenleaf Concepts, Inc.