World record ($100,000) prime number found?

Researchers may have turned up the 45th example of a Mersenne prime—a type of prime number rare enough that months or years of computerized searching are required to pick one out among the throngs of mere primes.

Details are still sketchy but the Great Internet Mersenne Prime Search (GIMPS) has announced on its Web site that a computer turned up a candidate Mersenne (pronounced mehr-SENN) prime on August 23. Checking began this week and should be completed by September 16.

If it checks out, the finding of the 45th Mersenne prime (MP) might qualify for a $100,000 prize offered by the Electronic Frontier Foundation for anyone who a prime number having at least 10 million digits. The 44th MP, discovered in September 2006 by two researchers at Central Missouri State University, clocked in at 9.808358 million digits.

Mersennse primes, named for 17th-century French smarty-pants monk Marin Mersenne (left), follow the formula 2^p – 1, where the power p is itself a prime number. (Commenters, don't hesitate to pounce on errors in my arithmetic.)

Take p=3:

2^3 – 1
= 8 – 1
= 7, which is prime

But not all p's yield the Mersenne variety.

Consider p=11:

2^11 – 1
= 2048 – 1
= 2047
= 23 * 89
(T4P = thanks for playing)

The 44th MP had p of 32,582,657.

People aren't hunting for Mersenne primes in order to prove anything about them, according to Mike Breen of the American Mathematical Society. "They're doing it because it's there, and it's an interesting challenge," he says. Math nerds also go ga-ga for really big numbers, as we all do I'm sure.

Here's a side note courtesy of Breen (to whom no errors of mine should be attributed): Mersenne primes are all associated with "perfect numbers," those such as 6 or 28 whose factors add up to themselves (or to double themselves if you include the number itself as a factor). E.g., the factors of 28 are 1, 2, 4, 7 and 14, which add up to 28.

There's a simple formula relating the two:

        Perfect Number = MP * 2^(p-1)

Take p=3 again:

(2^3 – 1) * (2^[3-1])
= 7 * 2^2
= 7 * 4
= 28

I leave the proof of the relationship to the reader.

Related ($): The new way to do pure math: experimentally

See also: "@Home" projects band together and proliferate 



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