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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
(QED)

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 

15 Comments

1. 1. SAJP2000 03:12 PM 8/28/08

   Any good psychic should be able to find the 45th MP rather quickly, no?
   
   
   
   Hehehe... a little joke...

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2. 2. Chaosqueued 03:54 PM 8/28/08

   I'm not much of a numbers person, but i was wondering if there was any thing to the fact that the first four Mersenne Primes yeild Mersenne Primes. M2 = 3, M3 = MM2 = 7, M7 = MMM2 = 127

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3. 3. ranibaron 06:30 PM 8/28/08

   You have an error in the sample formula:
   (2^3  1) * (2^[3-2]) should be (2^3  1) * (2^[3-1]) that is the 3-2 should be 3-1

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4. 4. ranibaron 06:30 PM 8/28/08

   You have an error in the sample formula:
   (2^3 – 1) * (2^[3-2]) should be (2^3 – 1) * (2^[3-1]) that is the 3-2 should be 3-1

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5. 5. trss 11:02 AM 8/29/08

   that was amazing! yeah, a small mistake pointed out by ranibaron. Chaosqueued may have a point there. nice!

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6. 6. oto81 04:59 PM 8/30/08

   Is 28 a factor of 28? if so, then shouldn't your sentence read "" (or to double themselves if you EXCLUDE the number itself as a factor)..."

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7. 7. trss in reply to oto81 09:17 PM 8/30/08

   i guess it is correct as it is. if 28 is considered as a factor of 28 then the sum of factors of 28 will be double itself which is 56.

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8. 8. JustinDoDrop 10:49 AM 8/31/08

   Wow dude, that is just way too cool. this dude is obviously very smart!
   http://www.useurl.us/17n

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9. 9. WRT 10:23 PM 8/31/08

   The aritcle states: "... 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."
   
   I believe that is FALSE. We're searching for Mersenne primes among the throngs of natural numbers. It's not the case that we are finding a lot of primes, and carefully trying to identify the Mersenne primes within that set.
   
   The Mersenne Primes are the easy primes to find, because there is a formula that predicts good candidate Mersenne Numbers, which must then be tested to verify whether they're prime or not. Even so, many candidates are not prime. It is all this verification that is so computationally intensive.
   

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10. 10. kapauldo 11:26 PM 8/31/08

    (linkback) Cool or Boring? 45th Mersenne Prime found, qualifies for $100,000 prize from EFF [VOTE] - http://www.thriveorfail.com/2fc9d

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11. 11. oto81 in reply to trss 09:39 PM 9/1/08

    Sorry, it is the words before the parenthetical. "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)." I think you mean "...add up to themselves (if you exclude the number itself as a factor) or double themselves (if you include..."

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12. 12. oto81 in reply to trss 09:44 PM 9/1/08

    ah, so it is a number whose factors (excluding the number itself) add up to the number itself or, if including the number itself as a factor, add up to double the number. got it, thanks.

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13. 13. hector2000 12:32 AM 9/3/08

    it is well known that prime numbers are infinite.
    My question--has anybody proven whether Mersenne primes are infinite or NOT?

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14. 14. hector2000 12:34 AM 9/3/08

    it is well known that primes are infinite
    My question-- has anybody proven whether MERSENNE primes are infinite or NOT?

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15. 15. cdcinks 11:58 AM 9/5/08

    I'm wondering why the EFF would be paying $100k for this. I know that large prime numbers are a key part of some pseudo-random number generators and some cryptography... is that why there is an interest in funding this research?

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