MathWorld Headline News

40th Mersenne Prime Announced

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

December 2, 2003--Almost exactly two years after the 39th largest Mersenne prime was discovered (MathWorld headline news: November 14, 2001, announcement; December 5, 2001, confirmation), 2^20996011 - 1, a number having 6,320,430 decimal digits, has been identified as a Mersenne prime, making it the largest such number as well as, by far, the largest known prime discovered to date. This news followed a November 17 report (MathWorld headline news: November 19, 2003) on the Great Internet Mersenne Prime Search (GIMPS) website announcing that a new Mersenne number passed the Lucas-Lehmer test, thus identifying it as a prime number.

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 (sequence A000043 in Neil Sloane's online encyclopedia of integer sequences). 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) 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 latest prime was flagged by GIMPS volunteer Michael Shafer, a 26-year-old volunteer in the Mersenne.org research project. Shafer used a Michigan State University lab PC and free software by George Woltman and Scott Kurowski as part of an international grid of 211,000 networked computers.

Shafer used an off-the-shelf 2 GHz Pentium 4 Dell Dimension PC running for 19 days to prove the number prime, a result that was subsequently confirmed using independent hardware and a different algorithm. According to a news release on the mersenne.org website, Shafer, a chemical engineering graduate student at Michigan State University, immediately performed a short victory dance, then called up his wife and friends to share the news of his discovery.

The primality testing algorithm used by GIMPS was developed in Mathematica by Dr. Richard Crandall, Director of the Center for Advanced Computation at Reed College in Portland, Oregon. For those curious to see the new prime in its full 6,320,430 digits of glory, the results of the short Mathematica calculation required to generate its decimal digits are available for download below. The properties of this prime behemoth can also be explored using Mathematica by downloading the notebook mersenne40.nb. If you do not own Mathematica, you may download a free copy of MathReader to view this file.

Interestingly, the sixth through 10th largest known primes have all been discovered in the year 2003, with the five largest Mersenne primes (discovered between 1997 and 2003) holding down the top five 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.