*MathWorld* Headline News

## 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),
^{24036583} - 1

Mersenne numbers are numbers of the form *M _{n}* =
2

^{n}- 1. For example,

*M*

_{7}= 2

^{7}- 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 *M*_{13466917} 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 |

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

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

GIMPS. "GIMPS Home Page: 41st Known Mersenne Prime Found!!" www.mersenne.org

mersenne.org. "Mersenne.org Project Discovers New Largest Known Prime
Number, 2^{24,036,583} - 1.
Project Leaders Believe $100,000 Award Within Reach.
http://mersenne.org/24036583.htm

Weisstein, E. W. "*MathWorld* Headline News: 40th Mersenne Prime Announced."
Dec.~2, 2003.
mathworld.wolfram.com/news/2003-12-02/mersenne

Woltman, G. "Mersenne: 41st Mersenne Prime Reported!!" Message to Mersenne Prime Mailing List. May 15, 2004.

Woltman, G. "It's Official." Message to Mersenne Prime Mailing List. May 29, 2004.