Ivars Peterson's MathTrek
June 7, 2004
Mersenne was certainly correct about the smaller numbers. For example, when n = 7, (27 1) = 127, which is a prime number. However, Mersenne could not have tested all the candidates. It turns out that he was wrong about two of them (for n = 67 and 257, the numbers are composite) and also missed a few (for n = 61, 89, and 107, the numbers are prime).
Nonetheless, it's Mersenne's name that's now associated with numbers of the form 2n 1. And the Great Internet Mersenne Prime Search (GIMPS) continues to unearth new Mersenne primes.
The latest entrythe 41st known Mersenne primewas discovered last month:
Written out in the full, the number consists of 7,235,733 decimal digits or, in binary notation, an unbroken string of 24,036,583 1s. It's now the largest known prime number.
Its discoverer is GIMPS participant Josh Findley, who used his home computer and software provided by George Woltman and Scott Kurowski to find the enormous prime. Findley's computer is one of more than 240,000 computers worldwide engaged in testing Mersenne numbers for primality. GIMPS volunteers are responsible for checking Mersenne numbers within specified ranges of exponents, whenever their computers would otherwise be idle.
The new champion prime greatly surpasses the previous record holder, 220,996,011 1, which has 6,320,430 decimal digits.
The latest find suggests that the first 10-million-digit prime may be within reach. Its discovery would net a prize of $100,000 from the Electronic Frontier Foundation.
GIMPS volunteers haven't yet tested every Mersenne number smaller than the current champion, so another Mersenne prime may yet lurk among the untested numbers. Only exponents less than 12,441,900 have all been tested at least once.
There's still a lot more testing to do!
Copyright 2004 by Ivars Peterson
2004. Mersenne project discovers new largest known prime number, 224,036,583 1. Great Internet Mersenne Prime Search (GIMPS) press release. May 28. Available at http://www.mersenne.org/24036583.htm.
Peterson, I. 2003. Megaprime champion. MAA Online (Dec. 8).
Weisstein, E.W. 2004. 41st Mersenne prime announced. MathWorld (June 1). Available at http://mathworld.wolfram.com/news/2004-06-01/mersenne/.
Information about the Great Internet Mersenne Prime Search (GIMPS) can be found at http://www.mersenne.org/.
For additional information about Mersenne primes, see http://www.utm.edu/research/primes/mersenne/ and http://mathworld.wolfram.com/MersennePrime.html.
A poster (with optional magnifier) showing all 7,235,733 decimal digits of the largest known prime number (and the 41st Mersenne prime) can be purchased from Perfectly Scientific at http://www.perfsci.com/novelties.htm.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.