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47th Mersenne Prime Confirmed
radiot88 writes to let us know that he heard a confirmation of the discovery of the 47th known Mersenne Prime on NPR's Science Friday (audio here). The new prime, 2^42,643,801 - 1, is actually smaller than the one discovered previously. It was "found by Odd Magnar Strindmo from Melhus, Norway. This prime is the second largest known prime number, a 'mere' 141,125 digits smaller than the Mersenne prime found last August. Odd is an IT professional whose computers have been working with GIMPS since 1996 testing over 1,400 candidates. This calculation took 29 days on a 3.0 GHz Intel Core2 processor. The prime was independently verified June 12th by Tony Reix of Bull SAS in Grenoble, France..."
It was found through the GIMPS (The Great Internet Mersenne Prime Search). The site http://www.mersenne.org/prime.htm is currently down.
The system used for this is GIMPS, the Great Internet Mersenne Prime Search. The system uses a distributed computing system using unused computing power in personal computers to search for various candidate primes. Computers do one of two things: Either trying to factor candidate Mersenne numbers or running a Lucas-Lehmer test on candidates without any small prime factors (the Lucas-Lehmer test is a special primality test for Mersenne numbers that is very fast). They use modular arithmetic and a variant of the Fast Fourier Transform to handle the multiplications which might otherwise become too difficult. The procedure is naturally a problem that can be made into a parallel processing problem like this since there are so many different candidate numbers to look at.
The summary doesn't mention but it is worth noting that the Lucas-Lehmer test allows one to check the primality of Mersenne numbers (numbers of the form 2^p-1, p prime) much faster than you can test the primality of generic numbers (or almost any other specialized form). Thus, for most of the last hundred years the largest primes known have been Mersenne primes. Currently the largest known prime is a Mersenne prime and the next 4 largest are also Mersenne primes. The GIMPS website - http://mersenne.org/ has a lot more details of both the math and software and explains how you can join in to help the project.
The historical reasons for caring about Mersenne Prime are twofold: First, Mersenne primes correspond to perfect numbers (numbers that are the sum of their positive less than the number. So for example, 6 has as proper divisors 1,2 and 3 and 1+2+3=6). The ancient Greeks were fascinated by perfect numbers but could not do much to understand them. Euclid showed that if one had a Mersenne prime one can construct an even perfect number. In particular, if 2^n-1 is prime then (2^n-1)*2^(n-1) is perfect. Almost 2000 years later, Euler showed that every even perfect number is of Euclid's form. Thus, investigating Mersenne primes tells us more about perfect numbers. The oldest unsolved problems in math are 1) are there any odd perfect numbers? and 2) are there infinitely many even perfect numbers? Thus, investigating Mersenne primes helps us get closer to solving one of the two oldest unsolved problems in mathematics.