That should read 2[sup]p[/sup] - 1. Beyond that, it’s self explanatory. Not all 2[sup]p[/sup] - 1 are prime, and in fact, most are not, but if it is, then it’s a Mersenne prime. As stated, p must be a prime number, but it need not be a Mersenne prime. As an example, 2[sup]3[/sup] - 1 = 7, which is prime. 7 is therefore a Mersenne prime.
There’s also a connection between Mersenne primes and even perfect numbers. A perfect number is a number such that, when you sum the divisors of the number, you get twice the original number. 6 is such a number, since the divisors sum to 1 + 2 + 3 + 6 = 12 (twice six).
An even number is perfect if and only if it is of the form 2[sup]p-1/sup, where 2[sup]p[/sup]-1 is a Mersenne prime.
For example, the first four Mersenne primes are 3, 7, 31, and 127. And so the first four (even) perfect numbers are 23 = 6, 47 = 28, 1631 = 496, and 64128 = 8128.
Nobody knows if there are an infinite number of Mersenne primes (or, equivalently, if there are an infinite number of even perfect numbers).
Nobody knows if there are any odd perfect numbers.
Since it was an MSU student whose computer discovered the number, there was an article here State News (Michigan State University’s Student Newspaper) about the discovery. There’s some nice little tidbits about the number itself in the sidebar.
One interesting quote from the article was the rate at which the program being used is discovering new primes:
There’s no real point (or special usefulness) to my knowledge. As far as I know, the history of perfect numbers is rooted deeply in numerology. Certain numbers were associated with certain “mystical” properties, and a perfect number was an interesting number, indeed. Perfect numbers have since been considered an interesting topic by mathematicians, but to my knowledge it doesn’t run any deeper than an ancient, interesting curiosity, really.
An interesting aside (and I’ll give the caveat that I’m no Biblical scholar, so I may be wrong, I’m just going by what I’ve read other places) is Genesis 32:14:
220 goats and 220 sheep were offered as gifts. There are (according to what I’ve read) many instances of numerology in the Bible, and this is apparently one of them.
A perfect number, remember, is one whose divisors sum to twice the original number. A similar notion is the idea of friendly (or amicable) numbers, also dating back to numerology. Two numbers x and y are said to be friendly if the sum of the divisors of x, and the sum of the divisors of y, both sum to x + y. 220 and 284 are such a pair: