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Math

Largest Prime Number Discovered – With More Than 23m Digits (mersenne.org) 117

Posted by msmash from the breakthrough dept.
chalsall writes: Persistence pays off. Jonathan Pace, a GIMPS volunteer for over 14 years, discovered the 50th known Mersenne prime, 2^77,232,917 -- 1 on December 26, 2017. The prime number is calculated by multiplying together 77,232,917 twos, and then subtracting one. It weighs in at 23,249,425 digits, becoming the largest prime number known to mankind. It bests the previous record prime, also discovered by GIMPS, by 910,807 digits. You can read a little more in the press release.
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Largest Prime Number Discovered – With More Than 23m Digits More

Largest Prime Number Discovered – With More Than 23m Digits

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  • I'll fine one right now (Score:1, Insightful)

    by Anonymous Coward

    2^98,435,672 -- 1

    This is easy. Where's my fookin medal?

    • Exercise for the interested maths nerd: Prove that if q is any even number, then 2^q - 1 is divisible by 3.

      • Re: (Score:1)

        by Anonymous Coward

        Fails if q = -2.

      • Re:I'll fine one right now (Score:5, Interesting)

        by RackinFrackin ( 152232 ) on Thursday January 04, 2018 @07:01PM (#55865781)

        Not a rigorous proof, but here's my favorite explanation:

        for any positive integer k, the binary representation of 2^k-1 consists of k 1's. If k is even, this is an even number of 1's lined up together. Since 3 is 11 in binary, you can divide 2^k-1 by 3 and get a quotient of the form 10101..01.

        e.g. 2^10 = 1111111111=11(101010101)

        • Nice argument!

          That was clearly too easy. Show that 2^98,435,672 - 1 is divisible by 5.

          • Nice argument!

            That was clearly too easy. Show that 2^98,435,672 - 1 is divisible by 5.

            It won't calculate on Excel, therefore its truth is unverifiable.

        • 2^98,435,672 - 1 is equal to (2^49217836 + 1)(2^49217836 - 1); it's a difference of squares.

          • 2^98,435,672 - 1 is equal to (2^49217836 + 1)(2^49217836 - 1); it's a difference of squares.

            But we're discussing primes. what's 4^2 - 3^2 ?

      • Re: (Score:3, Interesting)

        by ClickOnThis ( 137803 )

        Exercise for the interested maths nerd: Prove that if q is any even number, then 2^q - 1 is divisible by 3.

        Because q is even, we can write 2^q - 1 = [2^r -+1] * [2^r - 1], where 2r = q and r is a (positive) integer.

        Now consider the numbers 2^r - 1, 2^r, and 2^r + 1. One of these numbers must be divisible by 3. Clearly it is not 2^r because it only has factors that are powers of 2. Therefore either 2^r - 1 or 2^r + 1 must be divisible by 3. QED.

      • As an extension... every prime greater than 3 is of the form 6n+1 or
        6n -1 (wish I knew how to get the plus or minus sign into Slash comments)

        6n is obviously not prime (divisible by 6)

        6n+2, 6n+4 are divisible by 2

        6n+3 is divisible by 3

        which just leaves 6n+1 and 6n+5

        6n+5 is equivalent to 6m-1 (where m=n+1)

        so 6n+1 6n-1 are necessary but not sufficient conditions but do provide a quick & dirty test for primality for small - medium sized numbers [divisibility by 2 is easily checked; divisibility by 3

  • If GIMPS Can Find Such a Huge Prime (Score:5, Funny)

    by Anonymous Coward on Thursday January 04, 2018 @04:18PM (#55864921)

    Just think how big a prime PHOTOSHOPS could find!

  • Every time a discovery like this is made I find it very cool, but is there any real application or point to identifying a larger prime number? Or all of the GIMPS folks burning CPU cycles just for the fun of it?

    • Re:Application (Score:5, Informative)

      by kaoshin ( 110328 ) on Thursday January 04, 2018 @04:26PM (#55864943)
      From TFA:

      "At present there are few practical uses for this new large prime, prompting some to ask "why search for these large primes"? Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers. For seven more good reasons to search for large prime numbers, see here [utm.edu].

      • Re:Application (Score:4, Interesting)

        by thePsychologist ( 1062886 ) on Thursday January 04, 2018 @07:56PM (#55866089) Journal

        To clarify, these types of primes aren't useful for cryptography as they are much too large and not 'typical' primes.

        From a theoretical perspective they are quite interesting: they are in bijective correspondence with perfect numbers and no one knows whether there are infinitely many. For all we know, this one could theoretically be the last.

        • Re: (Score:3)

          by ezdiy ( 2717051 )
          IMO, the main practical advantage is that checking for MP is fast, so they're easier to search for. Working with MP number fields is fast too (owing to close-to-pow2 property). so they find use whenever any large prime would do, it's just a nice to have MP in there.

          A typical practical usage is PRNGs of girgantual periods (the period is typically the MP itself, or multiple of thereof) for HPC number crunching. The perfect number property indeed is a nice bonus in there, as it often leads to better k-distr

    • Re: (Score:3, Informative)

      by semper_statisticum ( 3983263 )

      The more prime numbers that are discovered, the more likely we are to be able to discover a pattern within an arbitrary base number set. The larger numbers are useful because we also want to make sure that the entire range is consistent, or in other words that any pattern, or lack of pattern, is the same across the entire set of numbers. There is always a benefit to trying to find patterns in number theory -- it's one of the coolest and most interesting fields in pure mathematics.

      • I have to wonder if looking for just Mersenne primes will reveal anything interesting about the primes in general. It seems unlikely to me.

        Having said that, finding a pattern in the Mersenne exponents would be very interesting indeed.

        • An entity which could encode a message in the distribution of Mersenne exponents would be very powerful indeed. It'd be even harder than encoding a message into physical constants which a hyper advanced civilisation may be able to do by creating new universes inside black holes.

    • think of all the dogecoin they could have mined!

    • Re: (Score:3, Informative)

      by JoshuaZ ( 1134087 )
      Primarily for the fun of it. There are some specific uses of large Mersenne primes in the Mersenne twister algorithm for generating pseudorandom numbers https://en.wikipedia.org/wiki/Mersenne_Twister [wikipedia.org], but in practice much, much smaller Mersenne primes are perfectly fine for that use, and indeed are much more practical. There are people who whenever you talk about large primes will claim they are useful for crypto, but that's not generally the case. The primes are too big for practical Diffie-Hellman (and th

    • Re: (Score:2)

      by guruevi ( 827432 )

      It's interesting for pure mathematics people. There are some minor applications for this although it's mostly theoretical.

      Once we get bigger computers that can hold these numbers, they may be used to prime a PRNG or a cryptographic constant, especially once quantum computing starts threatening the constants in the lower ranges. Quantum computers can't break cryptography, they just do it faster and for larger primes you still need more q-bits.

      Highly theoretical there may be some constant to the prime numbers

  • Headline Color? (Score:1)

    by Anonymous Coward
    Neat, but, what's with the colour of the headline?

  • Is this an indication that we're going to be getting more placed content and less user-voted content?

    I'm not saying that's a good or bad thing - I'm just wondering what this is... or maybe it's already been around a while and I just am not observant.

  • Headline writer is a boob (Score:4, Insightful)

    by FatRatBastard ( 7583 ) on Thursday January 04, 2018 @04:27PM (#55864951) Homepage

    Subject says it all.

    • Yes, they obviously mean largest known prime, not the largest prime there is. Headlines need to be short and frequently need to rely on some minimal context. Yes, the headline is hardly a "Headless body found in topless bar" quality but it was pretty clear from the context what was meant.

      • So 6 extra characters (known_) is apparently a bridge too far? Its not that the posted headline can be interpreted in different ways; it is outright wrong.

    • The phrase "largest prime number discovered" refers to the largest prime that has been discovered. I know English is a hard language, but it's not that hard.

      Having said that, pity the Slashdotter who needs "raising a number to the power of two" explained to them.

      • Re: (Score:2)

        by Jeremi ( 14640 )

        The phrase "largest prime number discovered" refers to the largest prime that has been discovered. I know English is a hard language, but it's not that hard.

        The problem is that the phrase is ambiguous.

        It could mean:

        The (largest prime number) has been discovered. -- i.e. there is a largest prime number and we just found it.

        or it could mean:

        The largest prime number [that we have so far] discovered [is two-to-the-whatever-minus-one].

        Presumably the headline writer mean to communicate the latter, but ended up mostly communicating the former. It's true that people familiar with the properties of prime number can probably

  • discovered the 50th known Mersenne prime, 2^77,232,917 -- 1 on December 26, 2017.

    I've done the math and 2^77,232,917 -- 1 is not prime. Although decrementing it by 2 to get 2^77,232,917 - 1 is indeed a prime.

  • Thats (Score:5, Funny)

    by JustOK ( 667959 ) on Thursday January 04, 2018 @04:53PM (#55865129) Journal
    That's amazing! I've got the same combination on my luggage!

    • Re: (Score:2)

      by dohzer ( 867770 )

      Never mind, the TSA staff have cut your lock off. They're going through your underwear as we speak.

      • Re: (Score:2)

        by JustOK ( 667959 )
        I've saved alot of money since I've realized the TSA would do stuff like that for free.

  • 2^77,232,917 + 1

    • Re: (Score:2)

      by pjt33 ( 739471 )

      Number Theory 101 exercise: why is 3 the only Mersenne number to be the smaller half of a prime pair?

      • Re: (Score:2)

        by Opyros ( 1153335 )
        For those who don't know the answer: two raised to any odd power then added to one is always divisible by three.

        • Re: (Score:2)

          by pjt33 ( 739471 )

          There's a simpler pigeonhole answer: one of (2^n - 1), 2^n, (2^n + 1) is divisible by 3. Clearly it's not 2^n. Therefore if (2^n - 1) is a prime other than 3, (2^n + 1) is the one which is divisible by 3.

    • That's not how prime numbers work.

  • Now for the important questions: Is it executable? And is it illegal?
    http://fatphil.org/maths/illeg... [fatphil.org]

  • I evaluated the full number using the mpmath Python library. It starts with:

    46733318335923109998833558556111...

    and ends with: ...1136582730618069762179071

    It took over an hour, but there are likely better ways to do it than I did, even with mpmath.

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