A Mersenne prime is defined as a prime of the form 2p−1, where p is any integer (and is always a prime itself). Early on, mathematicians believed that all numbers of this form were themselves primes (such as 3, 7, and 31), but in 1536 Hudalricus Regius proved that 211−1 = 2047 = 23 * 89 is not prime. Mersenne primes have a rich history dating back to Euclid’s Elements, in which the great Greek geometer (say that five times fast) proved that every perfect number — a number whose divisors less than itself sum to itself, like 6 = 1 + 2 + 3 — is related to a Mersenne prime in that its formula must be (2p-1)(2p−1), where p is some prime.
Funny story — in 1876, Edouard Lucas had shown, without finding factors, that M67 was in fact not prime as Mersenne had originally conjectured, but no factors were found for a quarter of a century, when Frank Nelson Cole, in a famous talk, walked — without speaking a single word — up to the blackboard, calculated M67 and multiplied 193,707,721 by 761,838,257,287, got the same number, and returned to his seat, to applause. Gotta love math audiences.
Above are two graphs. The first shows the number of Mersenne primes known to the mathematics community since 1603, when Pietro Cataldi correctly showed that 2p−1 was prime for p = 17, 19, bringing the known number of Mersenne primes to 7 and establishing modern interest. The graphics demonstrate swift advancement in both number theory techniques and computer analysis since the onset of the twentieth century. The most recently discovered Mersenne prime (which is, see below, the largest prime known) was discovered in January and made science news, but no such primes had been discovered since 2009, indicating either an apparent lull in discovery or, more likely, an increase in the complexity of the problem. The last fourteen Mersenne primes were discovered by the collaborative project GIMPS - Great Internet Mersenne Prime Search - which clocks out at 137 Tflops/sec (roughly 7,000 iPhones running in parallel) — and of those fourteen, eleven were the largest prime number known at the time of their discovery.
Interestingly, the ten largest prime numbers known to mathematics are themselves Mersenne primes. I say “interesting” because of the 1,622,441 prime numbers up to 25,964,951, for only 42 of them is 2p−1 also prime (that’s 0.0026%).
So, challenge, everyone! The first three Mersenne primes are 3, 7, and 31. Find the next Mersenne prime (without cheating :P) — this can be done numerically. Want an even bigger challenge? Find the fifth Mersenne prime — for this you might need a computer, unless you’re really good at arithmetic. An even bigger challenge, you say? Well, you already know the sixth and seventh Mersenne primes are M17 and M19. Find the eighth (Leonard Euler, that mathematician’s mathematician, verified this one…if you can do it without any sort of help, you should probably not be sitting at a computer reading our blog posts).
[CJH | sources: wikipedia.org, primes.utm.edu, mathworld.wolfram.com]