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Largest Twin Prime Yet Discovered

Posted by kdawson on from the Hardy-Littlewood-conjecture dept.

Chris Chiasson writes "The Twin Internet Prime Search and PrimeGrid have recently discovered the largest known twin prime. A twin prime is a pair of prime numbers separated by the integer two. The pair discovered on January 15th was 2003663613 * 2195,000 ± 1. The two primes are 58,711 digits long. The discoverer was Eric Vautier, from France."

42 of 160 comments (clear)

  1. Are you kidding? by greg_barton · · Score: 5, Funny
    A twin prime is a pair of prime numbers separated by the integer two.

    Are you kidding? Those are easy to find! Try getting two primes separated by the integer three...
    1. Re:Are you kidding? by EmagGeek · · Score: 5, Funny

      You mean like this?

      137

      The primes are 1 and 7, separated by the integer 3...

    2. Re:Are you kidding? by fredmosby · · Score: 3, Insightful

      How about 2 and 5.

    3. Re:Are you kidding? by Peter+Cooper · · Score: 4, Informative

      Sorry to take a dump on a cute joke with pedantry, but 1 isn't a prime.

    4. Re:Are you kidding? by proverbialcow · · Score: 4, Funny

      Very good. Now try finding two primes whose difference is 7.

      And when you're done with that, find two perfect cubes whose difference is also a perfect cube. I did this once, but there wasn't enough room in the margin to write the answer.

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    5. Re:Are you kidding? by cperciva · · Score: 4, Informative

      Now try finding two primes whose difference is 7.

      How about 5 and (-2)?

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    6. Re:Are you kidding? by Secret+Rabbit · · Score: 5, Informative

      To join this little debate (replying to you as I don't want to reply to two different people with the same post):

      Actually, if one considers 1 a prime problems end up happening e.g. inconsistencies with algebraic number theory (prime ideals) and elementary number theory. Basically, if you pop in 1, elementary number theory is fine (at least up to where I've studied it doesn't really matter aside from making some proofs more difficult than necessary). But, then some further developments like algebraic number theory start having problems, like the before mentioned inconsistency in the definition of a prime.

      Leaving 1 out as a prime makes the elementary number theoretic definition consistent with the algebraic number theoretic definition. Just thought I'd point that out as math is all about detail and consistency. And not having a consistent definition of a prime is a rather large f**k up as we all know how important primes are.

      So, although 1 has been considered a prime in the past, it does seem (keep in mind, I've looked through several libraries) that 1 has been dropped as a prime. Modern mathematics seems to have taken care of this discussion.

    7. Re:Are you kidding? by cperciva · · Score: 2, Informative

      Nice try.

      Somehow I'm not surprised to find that materials written for consumption by grade school students (and teachers) get this wrong. A prime element of an integral domain is a non-zero non-unit p such that if p divides ab, p divides either a or b (or both). The integers are an integral domain, and (-5) is a prime.

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    8. Re:Are you kidding? by stupid_is · · Score: 2, Informative
      Interesting - in my maths degree and at school (in the UK), we were taught that log(x) was base 10, and ln(x) was the natural log. Other ways of writing it would be to include the base as a subscript to the log(), which made it more obvious when doing those tedious exercises to convert the base.

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    9. Re:Are you kidding? by Record+Keeper · · Score: 5, Funny
      (for the record I don't treat 1 as a prime number)

      Noted.

    10. Re:Are you kidding? by fatphil · · Score: 2, Informative

      "Now try finding two primes whose difference is 7."

      4+w and 11+w in the Eisenstein Integers. (so w is the primitive cube root of unity)

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    11. Re:Are you kidding? by fatphil · · Score: 2, Informative

      Bollocks.

      2+w and 9+w.

      Arse.

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      Also FatPhil on SoylentNews, id 863
  2. Good example of a /. story. by Ninjaesque+One · · Score: 4, Insightful

    Succinct, on a subject undeniably nerdy, and mostly devoid of spelling mistakes. Also, not 'edited' by Zonk.

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    Ninjas and pirates. How piquant.
    1. Re:Good example of a /. story. by MagicM · · Score: 2, Funny

      Wouldn't that make it a bad example of a /. story?

      *rimshot*

  3. How is this meaningful? by JimMcc · · Score: 3, Interesting

    Seriously. I'm not a math major, etc. But I'm curious, is this of value? Other than of course as a curiosity.

    1. Re:How is this meaningful? by hamburger+lady · · Score: 2, Funny

      these numbers can totally come in useful in finding a cure for cancer.

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    2. Re:How is this meaningful? by 0rionx · · Score: 3, Informative

      This article is a pretty good summary of the reasons behind the search for large primes. Although finding a new large prime doesn't necessarily have any specific, short term "benefits", it serves to deepen our understanding of mathematics. As extremely large primes are of importance in cryptography, the ability to find and work with large primes has a great deal relevancy in IT, as well. The more we discover large primes the more we learn about ways to factor them quickly and efficiently.

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    3. Re:How is this meaningful? by TravisW · · Score: 5, Informative

      It depends on what you mean by "of value."

      At any rate, any particular pair of twin primes is unlikely* to be especially "significant." However, an important open problem in math is, "Do there exist infinitely many twin primes?" Experts think it's likely enough that the answer is yes that they've named that supposition "Twin Prime Conjecture," which indicates that those experts consider it definitely less than a theorem but much more than a wild guess.

      That the problem is so simply stated but remains unsolved is a testament to its difficulty (cf. Fermat's Last Theorem a.k.a. Wiles' Theorem). Hardy and Wright wrote to this effect: "The evidence, when examined in detail, appears to justify this conjecture, but the proof or disproof of conjectures of this type is at present beyond the resources of mathematics."

      *If the conjecture is false, that is, if there are only finitely many twin primes, certainly the largest pair is important.

      Incidentally, the "Pentium bug" was discovered when someone computed the reciprocals of two large (twin) primes and noticed an error after about 10 decimal paces.

      Twin Prime (Wikipedia)

    4. Re:How is this meaningful? by QuantumG · · Score: 3, Funny

      scoff all you want. You wouldn't believe the kinds of math that have been applied to gnome sequencing.. stuff that was discovered in completely different domains. That's the beauty of math.

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      How we know is more important than what we know.
    5. Re:How is this meaningful? by heinousjay · · Score: 3, Funny

      I usually just line the gnomes up by height.

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  4. Fun stuff by A+beautiful+mind · · Score: 2, Interesting

    Some people are very good at finding these primes. The now disposed record twin prime's finder was prof. Járai, whose lectures I attended.

    I find it interesting that the guy who works with insanely cool things like primes gave mind-numblingly boring lectures. He basically read his book out aloud. Some people are just very good at research and very bad at teaching.

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    1. Re:Fun stuff by DirePickle · · Score: 2, Funny
      I find it interesting that the guy who works with insanely cool things like primes gave mind-numblingly boring lectures.
      Only on Slashdot.
  5. Re:Don't seem too excited by odasnac · · Score: 5, Funny

    generally, yeah. most prime numbers are odd.


    ...

    i'm so sorry.

  6. Re:Huh? What? by tepples · · Score: 3, Interesting

    Actually, a twin prime is a pair of numbers n + 1 and n - 1 such that both are prime. For example, 41 and 43 are twin primes. Incidentally, if n is greater than 4, then n is always a multiple of 6; this is fairly easy to prove to yourself.

  7. Re:I am a math major... by Anonymous Coward · · Score: 5, Funny
    I am a math major (although I don't study prime numbers). This is totally, utterly useless, in a practical sense. Well, it might be useful in the field of CS, although I don't know enough about these project to know if any novel algorithms were used. It is sort of interesting though, because the twin prime conjecture (i.e. the statement that there are an infinite number of such pairs) is still unproven, so it's kind of cool to be able to say "Look, we found another pair!"

    One down, infinity more to go. Proof by enumeration, here we come...

  8. Re:Don't seem too excited by cperciva · · Score: 2, Funny

    most prime numbers are odd.

    Only on slashdot would the parent get moderated as "informative"...

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  9. Re:I am a math major... by AKAImBatman · · Score: 2, Funny
    This is totally, utterly useless, in a practical sense.

    Are you kidding me?!? I'm going to use that as my new encryption key! It will be like UBER-secure and take ten hundred billion, billion YEARS to guess!

    [...]

    Um... I wasn't supposed to tell you that, was I?
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  10. Re:Don't seem too excited by cgibbard · · Score: 3, Insightful

    Finding twin primes like this is mostly just an elaborate computational game which doesn't really tell much about the mathematical structure of twin primes. It doesn't help at all with knowing whether there are infinitely many or not, for example. The same goes for other searches for large primes.

    Also, if you're asking about real-world practical considerations, the primes used in practical work by comparison are tiny. Using such large primes for things like cryptography would be stupid for a number of reasons, not the least of which being that there are only so many known such primes out there, the size of your key would give it away. Personally, I don't know of any practical use for twin-primes or Mersenne primes, or any of the other classes of large primes being searched for.

    It's really more just for fun, like computing digits of pi. However, devising new ways to access large twin primes, for instance, results in improvements of our knowledge of them. It's those new theorems and algorithms which people might get excited about. Running a computer for hours or days or months to actually find the things is less interesting. ;)

  11. Minor correction by Zadaz · · Score: 5, Interesting

    "The discoverer was a computer in France, owned by Eric Vautier."

    I never felt like I should be allowed to take credit for what my screen saver does. Espcially since the whole point is that it does it when I'm not doing anything.

    'Course this will all be sorted out when computers can vote.

  12. Good for security. by r00t · · Score: 4, Funny

    Now we all know the best numbers to use for a PGP key.

  13. GMP by bellyjean · · Score: 2, Informative

    For the curious...

  14. Re:Don't seem too excited by Danny+Rathjens · · Score: 5, Funny
    most prime numbers are odd.
    Only on slashdot would the parent get moderated as "informative"...
    Do you know why 2 is odd?

    It's the only even prime number. :)
  15. Re:Huh? What? by XaXXon · · Score: 4, Informative

    Let's see if it really is fairly easy :)

    That gives us 5 other things to try:

    No odd numbers can be the base of a twin prime because adding or subtracting one leaves an even number which cannot be prime (except 2), so that knocks out
    6n+1, 6n+3, 6n+5.

    6n+2 and 6n+4.. why are those no good?

    6n+2 doesn't work because 6n is always a multiple of 3, adding 2 and then 1 (for the higher of the potential of the 2 twin primes) is also divisible by three, so it can never be a prime.

    6n+4 has the same problem, just on its lower possible twin prime.

    That took me longer to figure out that I'm happy with, but I think I got it :)

  16. No Biggest Prime: Proof by seawall · · Score: 2, Informative
    > Sometimes I get off the toilet and think I discovered the biggest prime...

    Most people here probably know this but:

    There is no biggest prime number and the proof is 2 sentences long.... here it is:

    Assume there is a largest prime P(n) and thus there is a finite list of all prime numbers: P(1), P(2), P(3),.....P(n). "*" here means multiply.

    Well then (P(1)*P(2)*...*P(n))+1 must be prime: whenever you divide that number by prime(s) you always have a 1 left over....but (P(1)*.....*P(n))+1 is obviously bigger than P(n) so our initial assumption of a largest prime number must be wrong. QED.

    One of the interesting things for mathematicians (or at least this ex-mathematician) is that you tweak the question just a little bit: "Is there a largest "twin prime"?" and heavy duty brains pound on the question for centuries with no answer. I have had NIGHTMARES over that one....which is one reason I am an ex-mathematician.

    Another funny thing about higher math is it has been defended as useless (Hardy: A Mathematicians Apology) but then three guys go and invent RSA and all of a sudden my privacy depends on the properties of prime numbers.

  17. Why call them twin primes... by sehlat · · Score: 5, Funny

    and not prime mates?

  18. To quote Fark by symbolset · · Score: 2, Funny

    This article is worthless without pictures.

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  19. power consumtions by AndyST · · Score: 2, Interesting

    It occurs to me that the power consumed for this kind of calculations is quite high. Back when I was doing seti@home, the classic one, they explicitly told people not to let computers running for the sole purpose of calculation, even asking them to turn them of when you guys in the US had a power crisis. There are people running farms of computers just for the fun of it. *sigh*

    seti, primes and stuff might be important, but I'd like to still have some power left to radio a reply to E.T.

  20. But... aren't all odd numbers prime ? by dargaud · · Score: 5, Funny

    Time for an old classic: How to prove that all odd numbers are prime?
    Well, this problem has different solutions whether you are a: Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction we have that all the odd integers are prime. Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error... Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime... Chemist: 3 is prime, 5 is prime... hey, let's publish! Modern physicist using renormalization: 3 is prime, 5 is prime, 7 is prime, 9 is ... 9/3 is prime, 11 is prime, 13 is prime, 15 is ... 15/3 is prime, 17 is prime, 19 is prime, 21 is ... 21/3 is prime... Quantum Physicist: All numbers are equally prime and non-prime until observed. Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student. Confused Undergraduate: Let p be any prime number larger than 2. Then p is not divisible by 2, so p is odd. QED Measure nontheorist: There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1). Cosmologist: 3 is prime, yes it is true.... Computer Scientist: 10 is prime, 11 is prime, 101 is prime... Programmer: 3 is prime, 5 is prime, 7 is prime, 9 will be fixed in the next release, ... C programmer: 03 is prime, 05 is prime, 07 is prime, 09 is really 011 which everyone knows is prime, ... BASIC programmer: What's a prime? COBOL programmer: What's an odd number? Windows programmer: 3 is prime. Wait... Mac programmer: Now why would anyone want to know about that? That's not user friendly. You don't worry about it, we'll take care of it for you. Bill Gates: 1. No one will ever need any more than 3. ZX-81 Computer Programmer: 3 is prime, Out of Memory. Pentium owner: 3 is prime, 5 is prime, 7 is prime, 8.9999978 is prime... GNU programmer: % prime
    usage: prime [-nV] [--quiet] [--silent] [--version] [-e script] --catenate --concatenate | c --create | d --diff --compare | r --append | t --list | u --update | x -extract --get [ --atime-preserve ] [ -b, --block-size N ] [ -B, --read-full-blocks ] [ -C, --directory DIR ] [--checkpoint ] [ -f, --file [HOSTNAME:]F ] [ --force-local ] [ -F, --info-script F --new-volume-script F ] [-G, --incremental ] [ -g, --listed-incremental F ] [ -h, --dereference ] [ -i, --ignore-zeros ] [ --ignore-failed-read ] [ -k, --keep-old-files ] [ -K, --starting-file F ] [ -l, --one-file-system ] [ -L, --tape-length N ] [ -m, --modification-time ] [ -M, --multi-volume ] [ -N, --after-date DATE, --newer DATE ] [ -o, --old-archive, --portability ] [ -O, --to-stdout ] [ -p, --same-permissions, --preserve-permissions ] [ -P, --absolute-paths ] [ --preserve ] [ -R, --record-number ] [ [-f script-file] [--expression=script] [--file=script-file] [file...]
    prime: you must specify exactly one of the r, c, t, x, or d options
    For more information, type "prime --help'' Unix programmer: 3 is prime, 5 is prime, 7 is prime, ...
    Segmentation fault, Core dumped. Computer programmer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 9 is prime, 9 is prime, 9 is ...
    Oops, let's try that again:
    3 is prime, 5 is prime, 7 is prime, 9 is ... 3 is prime, 5 is prime, 7 is prime, 9 is ... 3 is ...
    Um, right. Okay, how about this:
    3 is not prime, 5 is not prime, 7 is not prime, 9 is not prim

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  21. NO NO NO by Kjella · · Score: 2, Insightful

    Well then (P(1)*P(2)*...*P(n))+1 must be prime:

    No, no and even more no. Let's say my list of known primes is (3,5). 3*5+1 = 16 is not prime, all you've proven is that your list of primes is incomplete. It is only an existance theorem, and can not be used to find new primes.

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    1. Re:NO NO NO by Kjella · · Score: 2, Insightful

      Reread your formulation once again, and you claim you can list all primes less than p(n), which is different than the standard formulation of Euclid (he just says, take a list of known primes). But you're still wrong:

      2*3*5*7*11*13=30030
      30030+1=59*509

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    2. Re:NO NO NO by Orkan · · Score: 2, Informative

      You're right in a way, that method doesn't give you a prime in general. No one suggested that it did though. The proof was a proof by contradiction, i.e assume notX and generate a contradiction. Based on the assumption that there is a largest prime the procedure works fine. P(i) does not divide P(1)...P(n)+1 for any i in 1..n and by the assumption these are the only primes. The whole body of the proof is showing that "There is a maximum prime P(n)" => "There is a prime > P(n)" giving the contradiction. Therefore there is no maximum prime P(n) and so there must be an infinite number of them.

  22. yes yes yes yes by wpegden · · Score: 2, Insightful

    The poster is not wrong, his proof is correct. It is a proof by contradiction. He ASSUMES (for the sake of trying to find a contradiction), that there are finitely many primes P(1), ..., P(n). If there WERE only those finitely many primes, than the number P(1)*...*P(n)+1 WOULD be prime (because it's not divisible by any of them), which WOULD be a contradiction. Get it? Of course it's true that 2*3*5*7*11*13+1 may not be prime. But the poster didn't prove that P(1)*...*P(n)+1 is ALWAYS prime, he proved it was prime IF those are the only primes, which is enough to get a contradiction.

    Anyways, talking about what Euclid did is kind of irrelevant here (except from a historical perspective, of course). What he said wouldn't hold up in most math classes these days. Rather than doing an actual general proof, he says, "assume there are only 3 primes p,q,r. Then p*q*r+1 would also be prime, contradiction!" or something like that. Proof conventions have changed somewhat since then ;)

    Anyways, I guess this shows us that Slashdot's moderation system is no substitute for peer review in mathematics, even for really basic problems... surprise!