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Math Science

45th and 46th Mersenne Primes Confirmed 47

kahunak writes to alert us that GIMPS has announced that the 45th and 46th Mersenne primes have been confirmed. The EFF's $100,000 award, for the first prime over 10 million digits in length, will probably be claimed. (We discussed no. 45 when it was announced.)
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45th and 46th Mersenne Primes Confirmed

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  • Well, last week I discovered the prime number 37. It was only a matter of time before I discovered one greater than 10 million digits.

    Now these show-offs have gone ahead and spoiled it for the rest of us.
  • by RAMMS+EIN ( 578166 ) on Saturday September 13, 2008 @04:20PM (#24992909) Homepage Journal

    Not knowing why Mersenne primes matter, I looked it up on The Ultimate Source Of Truth [wikipedia.org]. From The Fine Article [wikipedia.org]:

    Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether there is a largest Mersenne prime, which would mean that the set of Mersenne primes is finite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.

    Mersenne primes are used in pseudorandom number generators such as Mersenne Twister and ParkMiller RNG.

    Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers.

    Mersenne numbers are very good test cases for the special number field sieve algorithm

    Out of those, I only knew about the connection with pseudorandom number generators, which I became interested in after writing my deadbeef random number generator [inglorion.net].

    • by l2718 ( 514756 ) on Saturday September 13, 2008 @05:11PM (#24993335)
      I think this tell us a lot more about the potential power of distributed computing than about prime numbers. While Mersenne primes are interesting to number theorists, we'll never find enough to do statistics on -- they are mostly of interests to pure mathematicians for reasons of curiosity. Random prime numbers of about 1024 bits are much more useful (and easier to find). On the other hand, if these was ever a problem we really needed to solve (protein-folding screensavers come to mind) then we now know how much computation power we can harness.
      • by kevinatilusa ( 620125 ) <kcostell@NOSpAm.gmail.com> on Sunday September 14, 2008 @12:35AM (#24995981)

        Knowledge of whether or not there are infinitely many Mersenne primes would probably not be interesting even to most pure mathematicians -- it's sort of a bizarre question that seems disconnected from the rest of mathematics. What would be interesting would be the actual methods used to prove this. In practice almost every question involving the existence/non-existence of certain types of primes is one we already know the answer to.

        The reason for this lies in the prime number theorem, which says that the proportion of numbers less than N which are prime is about 1/Log(N). Unless there's some compelling reason to believe otherwise, you can guess the answer to many problems involving primes by replacing them with a set randomly chosen with the same probability.

        For example, a randomly chosen number near 2^p-1 will be prime with probability about proportional to 1/p. Since the sum of 1/p diverges, we expect there to be infinitely many Mersenne primes (and can even guess their number, though this requires a bit more careful analysis to take care of the observation that Mersenne numbers don't have small prime factors, but this should only increase their number).

        The same trick allows us to guess the answer for twin primes (sum diverges, so there should be infinitely many) and Fermat primes (primes of the form 2^(2^n)+1 -- the sum converges, so there should be only finitely many). But none of these are really rigorous proofs, because they're all based on the fundamental assumption that the primes are somehow pseudorandom.

        Depending on the method of attack, a proof of the infinitude of Mersenne Primes may also shed light on how accurate or inaccurate the pseudorandomness assumption is. I would consider that to be a VERY interesting question.

  • by the_humeister ( 922869 ) on Saturday September 13, 2008 @05:16PM (#24993375)
    ... because they coincidentally correspond to two of Britney Spears's songs encoded as mp3 files at 128kb and the RIAA won't allow such copyright infringement! Double ouch!
    • Re: (Score:2, Insightful)

      by volxdragon ( 1297215 )

      ... because they coincidentally correspond to two of Britney Spears's songs encoded as mp3 files at 128kb and the RIAA won't allow such copyright infringement! Double ouch!

      If that's the case, no great loss, we wouldn't want to see (or hear) them anyway!

    • Re: (Score:3, Funny)

      by TheSpoom ( 715771 ) *

      This just in: Britney Spears is actually a weapon sent by aliens to enslave the Earth through hidden prime number telepathic messages.

      News at eleven.

    • I don't think even the RIAA is sick enough to enforce copyright infringement on Britney's songs. I mean anyone *THAT* sick to download em is capable of much more hideous actions and even the RIAA is scared of some things.
  • Well that suxs for Bruce [geekz.co.uk]. Now he has to change the combination on his luggage again.

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