The news of the discovery of a new, largest known prime broke about a week ago but I've only gotten around to writing about it here. It was of course a Mersenne prime discovered via GIMPS, the Great Internet Mersenne Prime Search. The number containing 22,338,618 digits is 2^74,207,281 - 1 where 74,207,281 itself must be prime of course. It is the 49th known Mersenne prime defined as a prime expressible in the form 2^p - 1 where p is prime. The first Mersenne primes are 3, 7, 31, and 127 corresponding to p values of 2, 3, 5, and 7 respectively. Note that p being prime is not sufficient to ensure that 2^p - 1 will be prime. As a counter example take p=11. The resulting number 2^11 - 1 = 2047 = 23 x 89 is not prime. Here are links to some more interesting information about Mersenne primes:
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