MathWorld Headline News

43rd Mersenne Prime Found

By Eric W. Weisstein

December 25, 2005--Less than a year after the 42nd Mersenne prime was reported (MathWorld headline news: February 26, 2005), the Great Internet Mersenne Prime Search (GIMPS) project has discovered the 43rd known Mersenne prime. The candidate prime was flagged prime by Drs. Curtis Cooper and Steven Boone, professors at Central Missouri State University. Additional details can be found in the Mersenne.org press release.

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. Mersenne primes are Mersenne numbers that are also prime, i.e., have no factors other than 1 and themselves. So, since the number 127 is prime and is a Mersenne number, it is a Mersenne prime.

The new Mersenne prime is 2^30,402,457 - 1 = 31541647561884608093...11134297411652943871 (where the ellipsis indicates that several million intervening digits have been omitted for conciseness) and has a whopping total of 9,152,052 decimal digits. It is therefore not only the largest known Mersenne prime, but also the largest known prime of any kind. In fact, there is a particularly efficient and, more importantly, deterministic primality test for Mersenne numbers known as the Lucas-Lehmer test. The efficiency of this test combined with the high historical profile of the Mersenne numbers thus accounts for the fact that the four largest known primes are all Mersenne primes.

For those curious to see the new prime in its full 9,152,052 digits of glory, the results of a short Mathematica calculation generating its decimal digits are available by downloading the notebook mersenne43.nb. If you do not own Mathematica, you can download a free trial version to view this file. A poster featuring all 9.1 million digits of the new prime (displayed in an extremely small point size) created by Richard Crandall, discoverer of the advanced transform algorithm used by the GIMPS program, is (or will shortly be) available from Perfectly Scientific.

The nine largest known Mersenne primes (including the latest) have all been discovered by GIMPS, which is a distributed computing project being undertaken by an international collaboration of volunteers. As of December, 2005, 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.

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). However, note that the region between the 39th and 40th known Mersenne primes has not been completely searched, so while the 40th number listed is the 40th Mersenne prime discovered, it is not yet known if M_20,996,011 is actually the 40th Mersenne prime.