### Now, That's a Number!

Just writing a number with more than ten million digits would be hard enough, but finding such a number with a specific property is quite an accomplishment. The Great Internet Mersenne Prime Search (GIMPS) announced on September 15, 2008, that two ten million digit Mersenne primes were discovered by a group of computers running collaborative software that breaks a large problem into smaller pieces [1-3]. The GIMPS virtual supercomputer on which these numbers were found is a network of 100,000 computers that has a combined throughput of nearly 30 teraflops [4-5]. The larger of these two numbers, which are 11,185,272 and 12,978,189 digits long, qualifies for a $100,000 award posted by the Electronic Frontier Foundation (EFF), of which I am a member. The EFF encourages collaborative activities in our internet age, and it advocates privacy and fair use rights in this age of easy electronic communication.

Mersenne primes are numbers M that are prime and satisfy the following condition,

M = 2^{n} - 1

in which n is an integer. Not all Mersenne numbers M are prime, as the simple example of (2^{4} - 1) proves. The two large Mersenne primes just found, (2^{37,156,667} - 1) and (2^{43,112,609}
- 1), are tentatively the 45th and 46th Mersenne primes. All Mersenne
primes from the 35th and above have been discovered by GIMPS, which has
been running since 1996. Since the GIMPS search is not sequential,
there may be other Mersenne primes found between those known. In fact,
the lesser of the two numbers was discovered after the larger. The last
Mersenne prime to be found without the use of an electronic computer
was (2^{127} - 1), discovered in 1876 by the mathematician, Édouard Lucas. Lucas worked for nineteen years on this task, aided by some essential mathematical tricks,
some of which he devised. Lucas began his test of this Mersenne prime
at age fifteen! It wasn't until 1952 that the next Mersenne prime was
discovered, this one on a computer.

The 12,978,189 digit prime was discovered by a computer at UCLA, so UCLA will receive half of the $100,000 prize. $25,000 of the remainder will be given to charity, and the other $25,000 divided among the GIMPS collaboration. How hard will it be to find a 100,000,000 digit prime? The award in that case will be $150,000, but I think a cent per digit (a million dollars) sounds like a fairer amount.

References:

1. J.R. Minkel, "World record ($100,000) prime number found?" (Scientific American Blog, August 28, 2008).

2. Huge new prime number discovered (BBC News, September 28, 2008).

3. Jenny Huntington, "Largest Mersenne Prime Number Discovered" (eFluxMedia, September 28, 2008).

4. The Great Internet Mersenne Prime Search (GIMPS).

5. Great Internet Mersenne Prime Search (Wikipedia).