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## 41st Mersenne Prime Announced

### By Eric W. Weisstein

June 1, 2004--Less than a half year after the 40th largest Mersenne prime was discovered (MathWorld headline news: November 19, 2003 announcement; December 2, 2003, confirmation), 224036583 - 1, a number having 7,235,733 decimal digits, has been verified as a Mersenne prime, making it the largest such number as well as the largest known prime discovered to date. This news follows a May 15 email message and report 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 Mn = 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 several previously known Mersenne primes has not been completely searched, so it is not known whether M13466917 is actually the 39th Mersenne prime.

The seven largest known Mersenne primes (including the latest) have been discovered by an international collaboration of GIMPS volunteers. The latest prime was flagged by GIMPS volunteer Josh Findley, a five-year volunteer in the research project. Josh's calculation took just over two weeks on a 2.4 GHz Pentium 4 computer, and was verified by Tony Reix with a five-day computation and independently by Jeff Gilchrist with an 11-day computation.

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 7,235,733 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 mersenne41.nb. If you do not own Mathematica, you can download a free trial version to view this file.

file format file size
mersenne41.txt plain text 7.4 MB
mersenne41.zip zip compressed 3.5 MB

References

Caldwell, C. K. "The Largest Known Primes." www.utm.edu/research/primes/largest.html

GIMPS: The Great Internet Mersenne Prime Search. www.mersenne.org