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Education Science

There Are Infinitely Many Prime Twins 479

Posted by michael from the so-the-theory-goes dept.
fustflum writes "R. F. Arenstorf from Vanderbilt University has presented a 38-page possible proof of the twin-prime conjecture using methods from classical analytic number theory. The paper is on arxiv.org and is freely available to the public. Twin primes are pairs of primes where both p and p + 2 are prime. "It is conjectured that there are an infinite number of twin primes ... but proving this remains one of the most elusive open problems in number theory." More information about twin primes can be found on Mathworld."
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There Are Infinitely Many Prime Twins More

There Are Infinitely Many Prime Twins

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  • Number theory (Score:5, Funny)

    by PHP Wolf ( 629571 ) <doublecompileNO@SPAMgmail.com> on Friday May 28, 2004 @05:51PM (#9281759) Homepage
    but proving this remains one of the most elusive open problems in number theory

    I think we all know the most elusive open problem in number theory is "How many licks does it take to get to the center of a tootsie pop?"

  • One smart dude (Score:5, Informative)

    by overbyj ( 696078 ) on Friday May 28, 2004 @05:52PM (#9281760)
    I was a grad student at Vanderbilt in a different department but I had some friends in math that really knew this guy. Needless to say, this guy is brilliant. I don't really know much about his work but honestly, I am surprised it took him as long as it did to do this.

    Score another for number theory thanks to this dude.

    • Re:One smart dude (Score:5, Insightful)

      by fatphil ( 181876 ) on Friday May 28, 2004 @06:11PM (#9281898) Homepage
      Slow down!
      It's not been reviewed yet.

      I'm waiting until Granville, Odlyzko, Mihailescu, or someone similar gives it the thumbs up.
      However, it's not obvious tosh, and therefore if it does have flaws it may well be correctible, or at least provide new insight.

      The guy certainly _was_ brilliant, but given that he started his peak in the mid-60s, there's no guarantee he's still at it.

      FP.

    • He makes a mistake... (Score:5, Funny)

      by b0r0din ( 304712 ) on Friday May 28, 2004 @06:27PM (#9281989)
      Look on page 27. He's trying to integrate homeomorphic convergence using a Baxter-Bates supermodality, which Krause clearly explained is impossible for T(s) in a non-linear progression.

      Ok, just fuckin with ya. My mind wandered after I saw the word 'Abstract.' ;)
      • I wonder if it's just me, or did a horde of really scary math geeks just mod that up? I sure didn't get it ;)

      • Re:He makes a mistake... (Score:5, Funny)

        by Anonymous Coward on Friday May 28, 2004 @07:19PM (#9282317)
        Yeah, I see a lot of people attempting to integrate homophobic conformance using Master-Bates supermoodality, which Krauds exploded as impenetrable for T/bag in a non-lesbian prostation.

  • This is why mathematicians are soooo popular. (Score:4, Funny)

    by The I Shing ( 700142 ) * on Friday May 28, 2004 @05:52PM (#9281765) Journal
    This stuff is so fascinating that I'm just sure I'll be the life of the party when I start talking about it!

    "You know, mathematicians theorize that there's an infinite number of prime twins, and... hey, where are you going?"

  • Old news (Score:5, Funny)

    by Hamster Of Death ( 413544 ) on Friday May 28, 2004 @05:52PM (#9281766)
    Glancing at my list of twin primes I can see it's infinite.

  • I have a better proof (Score:5, Funny)

    by Anonymous Coward on Friday May 28, 2004 @05:52PM (#9281771)
    but it hit /.'s maximum post size limit :(
      1. Given: There are infinitely many primes.
      2. Given: A certain positive percentage of primes differ by two.
      3. Given: Infinity times any positive number is infinity.
      4. Therefore: There are infinitely many primes that differ by two.
      That's my story and I'm stickin' to it.

      (Spot the logical error and you win a cookie!)

      • Re:I have a better proof, and it fits (Score:4, Insightful)

        by hoggoth ( 414195 ) on Friday May 28, 2004 @06:37PM (#9282057) Journal
        > 2. Given: A certain positive percentage of primes differ by two.

        Not necessarily true. It's equally possible that a certain finite number of primes differ by two, not an infinite percentage of primes.

        Give me my cookie now.

        • Re:I have a better proof, and it fits (Score:5, Informative)

          by SashaM ( 520334 ) <msasha&gmail,com> on Friday May 28, 2004 @07:05PM (#9282240) Homepage

          It's equally possible that a certain finite number of primes differ by two, not an infinite percentage of primes.

          Talking about infinite percentages is meaningless. Think about this question - what percentage of all natural numbers are even? On the one hand, it seems that since every second number is even, there would be 50%, right? But what if I pair each and every natural number to an even number so that two different numbers are paired to different even numbers (a one-to-one map)? Would that mean that 100% of all natural numbers are even? But it is done easily - I would pair each number n to 2*n.

          You could try and wiggle out of this problem by defining the infinite percentage to be the limit of the normal percentage until N when N goes to infinity. This would work for some sets, like the even numbers and would even give you a seemingly reasonable answer - 50%. But then consider this question - what percentage of all natural numbers are powers of 2 by this definition? I'll leave that as an exercise to the reader :-)

          See Cardinality [wikipedia.org]

        • Re:I have a better proof, and it fits (Score:4, Funny)

          by Atario ( 673917 ) on Friday May 28, 2004 @07:55PM (#9282473) Homepage
          Ding ding ding! We have a winnah!

          You will find your cookie on your hard drive, assuming you're logged in to Slashdot.
      • I know every one said "2" but "2" is true (it is a truism). The error is in "3".

        3. Given: Infinity times any positive number is infinity.

        This is not true. If you multiply an infinitely small number by an infinately large number, some times you get a finite number.

      • Re:I have a better proof, and it fits (Score:4, Interesting)

        by cubic6 ( 650758 ) <tom@NosPaM.losthalo.org> on Friday May 28, 2004 @06:52PM (#9282164) Homepage
        You can't assume that a certain positive percentage of *all* primes differ by two as stated in number two, because that's an analogous statement to what you're trying to prove.

        Say I have an infinite number of socks. All are white, except 3, which are grey. I have a positive percentage of grey socks, but that doesn't mean anything since that percentage is infinitessimal. It will be infinitessimal for any number of grey socks, so you can't say that you are assumed have a positive percentage of grey socks *unless* you have an infinite number of grey socks, and that's a tautological argument.

        Chocolate chip, please ;)
  • It is called "Prime Twins", they look the same!

    Good God. They look so god-damned like the same person... I would say to them, "you want ice-cream cone?", both of them say yes. How in the hell?

  • Can someone buy the editors a dictionary (Score:3, Funny)

    by Timesprout ( 579035 ) on Friday May 28, 2004 @05:54PM (#9281795)
    possible proof of the twin-prime conjecture

    The words possible and conjecture appear above. Where does the definitive statement "There Are Infinitely Many Prime Twins" come into it? Have the the /. editors secretly managed to prove this theory before posting it ?

    • Peer Review (Score:5, Informative)

      by Kozar_The_Malignant ( 738483 ) on Friday May 28, 2004 @06:43PM (#9282092)
      The author is saying, in effect, "I think I have a proof here. Have at it." His esteemed colleagues, including jealous backstabbers, hacks who have failed at the same problem, and a relatively small number of really first rate mathemeticians will try to show he is wrong. Consensus will emerge one way or another. The editors are, I'm sure, simply offerring the collective genius of /. a change to join the fray.

  • 38 pages? (Score:5, Funny)

    by Wakkow ( 52585 ) * on Friday May 28, 2004 @05:57PM (#9281808) Homepage
    They should have put it in 37 pages..

    • Re:38 pages? (Score:2, Funny)

      by Anonymous Coward
      They should have put it in 37 pages..

      Yes 37 is prime, but 41 is the nearest *twin* prime (with 43). So they should add 3 pages.

  • Well, one thing's for sure.. (Score:5, Funny)

    by robbo ( 4388 ) <slashdot@@@simra...net> on Friday May 28, 2004 @05:59PM (#9281818)
    they're all odd.

    (Waiting for my spot in the math hall of fame)

  • Prime Arithmetic Progression also in the news (Score:5, Interesting)

    by micha2305 ( 769447 ) on Friday May 28, 2004 @06:00PM (#9281828)
    Something I read in Science [sciencemag.org] the other day: There's a new proof in review that there are infintely many sequences such as 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 -- primes that differ by a constant offset. See also Mathworld [wolfram.com].
    • There's an old proof that there can't be an arithmetic progression all of whose members are prime. Such a progression is of the form f(x) = ax + b, and gives a composite number for all values of x which are a multiple of b (or, indeed, which aren't coprime with b). So since your link requires a subscription, could you please explain what the theorem actually says?

      • Re:Prime Arithmetic Progression also in the news (Score:5, Informative)

        by aardvarkjoe ( 156801 ) on Friday May 28, 2004 @06:35PM (#9282046)
        Quoting directly from the linked article:
        An arithmetic progression of primes is a set of primes of the form p1 + kd for fixed p1 and d and consecutive k, i.e., {p1, p1 + d, p1 + 2d, ...}. For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.


        In a recently published in preprint, Green and Tao (2004) use an important result known as Szemerédi's theorem in combination with recent work by Goldston and Yildirim, a clever "transference principle," and 48 pages of dense and technical mathematics, to apparently establish the fundamental theorem that the prime numbers do contain arithmetic progressions of length k for all k (Weisstein 2004).

        Take it for what it's worth. This stuff is way over my head.

  • Calm down, boys ... (Score:5, Funny)

    by RealAlaskan ( 576404 ) on Friday May 28, 2004 @06:02PM (#9281842) Homepage Journal
    That's ``twin primes'', not ``prime twins''. So, no, there is not an infinite supply of hot double dates.

  • Alien (Score:4, Funny)

    by Juiblex ( 561985 ) on Friday May 28, 2004 @06:03PM (#9281845)
    In what Alien language is the article written???

  • twins (Score:5, Funny)

    by sacrilicious ( 316896 ) <qbgfynfu.opt@recursor.net> on Friday May 28, 2004 @06:08PM (#9281884) Homepage
    Twin primes are pairs of primes where both p and p + 2 are prime.

    Even rarer are those pairs of primes known as the "conjoined twin" primes: those of the form p and p+1. Not many examples are known, but perhaps an infinite number are waiting to be discovered.

    • There are no "conjoined twin" primes other than 2 and 3. Rather trivial proof follows:
      • Assume p and p+1 to be primes, and p>2
      • Since p is prime and greater than 2, it does not have 2 as a factor, therefore it is odd
      • Since p is odd, p % 2 = 1
      • Since p % 2 = 1, (p + 1) % 2 = 0
      • Therefore (p + 1) is even, therefore (p + 1) has 2 as a factor, therefore (p + 1) is not prime
      • Therefore by contradiction, no conjoined twin primes exist other than (2,3)
  • If all else fails, use induction to prove?

    Boy, all those foundations of computer science courses I took are really paying off. :|

  • Ugh... math... (Score:3, Funny)

    by gwoodrow ( 753388 ) on Friday May 28, 2004 @06:14PM (#9281909)
    I tried to read through some of the paper and math websites... and I was suddenly reminded why the diploma that will be handed to me at the end of the summer will say:
    "Steven Gregory Woods... ENGLISH major"
    Hopefully, math will turn out to be just a fad :)
  • One thing that turned me off about math is the insistence in honoring these long dead people. That's focusing more on the discoverer than the discovery. I'm sure a reasonably intelligent person (like most of slashdot readers) could discover "Euclid's" infinitude of primes theorem. We probably would have if we didn't get in class along with the admonition to respect our elders. Take a patent if you want, Arenstorf. But don't insist people centuries from now worship you over the discovery.

  • This took 20 years (Score:5, Informative)

    by ortholattice ( 175065 ) on Friday May 28, 2004 @06:25PM (#9281977)
    Interesting quote from the paper (p. 3 of the PDF file):
    This work is the outcome of about twenty years of "on and off" search and research on this and the related binary Goldbach problem; in the interim having been lured onto various misleading paths or frustrated by (for me) insurmountable difficulties, before ultimately recognizing and constructing a workable approach.
  • I'm not sure I understand why this is so hard to figure out.

    Assuming that there are an infinite number of numbers (always n+1) then doesn't this have to be the case?

    • Re:Can someone give me the math here? (Score:3, Informative)

      by Anonymous Coward
      well, the fact that there are an infinite number of primes does not automatically mean that after some point there will exist a p with a twin prime. Say for example, after some such point all the primes are >2 apart, then is it not the case that there will be no more twin primes after this, even if there are an infinite number of prime numbers? I dunno, maybe it's too late. Anyway, the article is a "proof of the twin-prime conjecture". It was the slashdot editors that added the infinite number of twin p
      • Anyway, the article is a "proof of the twin-prime conjecture". It was the slashdot editors that added the infinite number of twin primes.

        No, the twin prime conjecture is that there are infinitely many twin primes, and the title was lifted directly from the paper. Are we now blaming the editors for correctness?
    • There are an infinite number of numbers, but there aren't an infinite number of pairs of primes p and p+3. (There is obviously only one such pair, 2 and 5.) So it's not trivial that there's not something which prevents there from being any further twin primes.

  • Although they are frequently confused, this conjecture has no bearing on so-called "Wonder Twin" primes, in which the p is in the shape of a polar bear and p+2 is in the form of an ice ladder.

  • Other Number Theory Tricks? (Score:4, Interesting)

    by CoolGuySteve ( 264277 ) on Friday May 28, 2004 @06:31PM (#9282017)
    The sum of any two consecutive odd numbers is divisible by 4. I misread a question in first year and proved it.

    I thought that was fairly neat, it makes me the life of the party when I tell it to people. (Well, not really. Depressing.) Does anyone know any other little tricks like that?

    • Re:Other Number Theory Tricks? (Score:2, Insightful)

      by Anonymous Coward
      Well, duh -- (2n-1) + (2n+1) = 4n. ;^)

      Here's my neat math trick: Take a multiplication table and go down the diagonal with all the perfect squares. Take one step northeast or southwest on the grid, and the new number is always one less than the one you came from.

      The only person I told this to, however, pretty much replied "Well, duh -- (n-1)(n+1) = n^2 - 1."

    • Re:Other Number Theory Tricks? (Score:4, Funny)

      by Jim Starx ( 752545 ) <JStarxNO@SPAMgmail.com> on Friday May 28, 2004 @08:40PM (#9282665)
      Here's an interesting one, this is guarenteed to piss off any math student that doesn't get it.

      if a=b, then:

      a^2=ab
      a^2-b^2=ab-b^2
      (a-b)(a+b)=b(a-b)
      a+b=b

      substitute in the original a=b equation

      2a=a
      2=1

      wtf? So where's the error? :)

      • Re:Other Number Theory Tricks? (Score:5, Informative)

        by Sycraft-fu ( 314770 ) on Saturday May 29, 2004 @12:11AM (#9283409)
        "So where's the error?"

        I'm guessing that's a rhetorical question, but the error is you divide by zero. On line three you are actually are showing 0=0 since anything minus itself is zero and anything times 0 is 0. You then try to divide out (a-b), which is zero, and can't be done.

        I can see this fooling people who aren't good at math but probably not math students. It's not like I ever got very far in math, and the problem is easy to spot.
  • Just out of curiousity, is there a practical reason to prove the existance of infinite numbers of twin primes? Or is this purely a matter of curiousity?

    • Re:Interesting, but what's the practical value? (Score:2, Insightful)

      by Anonymous Coward
      OK, so this is useless in itself, but you might consider this a motivation for studying number theory more generally.

      Products of two distinct prime numbers are significantly easier to factor when those primes are "near" each other. Therefore information about how primes are relatively distributed is useful.

      Of course, as I said before, this particular result isn't particularly helpful for cryptographic purposes, but you get the idea.

      No-one knows what mathematics will be 'applicable' in the future. Who

  • Infinite number of prime twins yes, but... (Score:3, Funny)

    by StandardCell ( 589682 ) on Friday May 28, 2004 @06:56PM (#9282180)
    There are still only four lights...
  • This is the best thing that has happened to mathematics research since the proof of Fermat's Last Theorem.
  • I think the pentium fdiv bug was revealed by some cat who was trying to prove that the series of of 1 / the gap between double primes converged. It might be the same guy.

    Just a little dorky computer math nerd trivia.

  • From the article
    This work is the outcome of about twenty years of "on and off" search and research on this and the related binary Goldbach problem; in the interim having been lured onto various misleading paths or frustrated by (for me) insurmountable difficulties, before ultimately recognizing and constructing a workable approach.
    I am inexplicably hyped about this. And I'd love to see a proof of Goldbach's conjecture in my lifetime.

    In the mode of some car-insurance commercial running in the US, I ran into my wife's office and said, "I've got great news!". Somehow, she didn't share my enthusiasm.

    When I was in high-school in 1978, my math teacher, Alan Crokall (sp?) gave me the programing/math assignment of either proving Goldbach's Conjucture or finding a counter example. He later explained that he wanted me to find the counter example so that it could be called "Goldberg's rejecture of Goldbach's conjecture".

    And you can find out about Goldbach's conjecture [wolfram.com] if you don't already know what it is.

  • Obvious Generalization (Score:4, Funny)

    by geordieboy ( 515166 ) on Friday May 28, 2004 @07:06PM (#9282245)
    I propose the geordieboy conjecture:

    There are an infinite number of prime n-pairs, where
    an n-pair is a pair of prime integers (p,p+n).

    I also propose geordieboy's second conjecture:

    There are an infinite number of prime tuples, where a prime
    tuple is a set of prime integers of the form (p+a,p+b,p+c,...)
    where (a,b,c,...) is a set of any integers of your choosing.

    Get stuck in you poor bastards!

  • amazing if it's true (Score:5, Interesting)

    by cancerward ( 103910 ) on Friday May 28, 2004 @07:08PM (#9282258) Journal
    The author received his doctorate 48 years ago. [ams.org] According to MathSciNet [ams.org] his first paper was in 1963, and his most recent in 1993.

    If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture. [temple.edu]

    You can follow discussions [google.com] on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. [maa.org] Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's [arxiv.org] along with almost-proofs like Castro-Mahecha's [soton.ac.uk] and Dunwoody's. [cnn.com]

  • What about prime triplets? (Score:3, Interesting)

    by MBraynard ( 653724 ) on Friday May 28, 2004 @07:22PM (#9282334) Journal
    3, 5, 7?

    Or prime siblings that are seperated by numbers other than 2?

    Just seems silly. I mean, they all probably exist in infinity.

    • Re:What about prime triplets? (Score:5, Interesting)

      by Sigma 7 ( 266129 ) on Friday May 28, 2004 @08:07PM (#9282518)
      3, 5, 7?
      There is only one set of prime triplets where the numbers are seperated by 2. There are no other triplets because at least one number in that triplet is a multiple of 3. (The numbers being X, X+2, and X+4. Using modular arithmtic to cap the additives would therefore require all numbers of the set X, X+2 and X+1 to not be a multiple of three, which isn't really possible because of how Integer numers work.)

      Or prime siblings that are seperated by numbers other than 2?
      To find an infinite number of prime siblings, you first need to find an appropriate set of numbers. To cut down on processing time, you should note that these numbers are seperated by 6, or a multiple thereof.

  • Interesting (Score:3, Interesting)

    by OneIsNotPrime ( 609963 ) on Friday May 28, 2004 @07:57PM (#9282477)
    Interestingly, it can be proven that there is a series of n consecutive composite (nonprime) numbers for ANY number n! This means there is some sequence of 10 trillion nonprime numbers. It seems almost contradictory to the infinicy of primes (though it is not).

    From http://www.fortunecity.com/emachines/e11/86/touris t2b.html -

    At the same time, it's relatively easy to prove that consecutive primes can be as far apart as anyone would want. The sequence of numbers n! + 2, n! +3, n! ..... . n! + n shows this conjecture must be true. The number n!, where n = 5, for example, has a value of 1 x 2 x 3 x 4 x 5, or 120. In the general case, n! + 2 is evenly divisible by 2, n! + 3 is evenly divisible by 3, and so on. Finally, n! + n is evenly divisible by n. Therefore, all the numbers in the sequence are composite. The sequence can be made arbitrarily long by picking a sufficiently large number n.

  • My way of viewing primes... (Score:3, Interesting)

    by Slur ( 61510 ) on Saturday May 29, 2004 @12:50AM (#9283590) Homepage Journal
    ...is probably not original, so maybe you can point me to something that conceives it exactly as I do.

    I see each prime number as the first integer in an infinite series of its multiples. I envision a line of infinite length, where each point on the line represents a number from 1 to infinity. For each prime number (beginning with 2) you place an X on the line where every multiple of that prime number falls. So for 2, you mark off every even number from 2 to infinity. Then for 3, every multiple of 3, and so on. Following this procedure in order, all you have to do to find any prime number is just locate the first unmarked integer on the line.

    If only it were possible to represent this abstract line inside a computer, all primes could be instantly located. Of course the marking-off part would take forever. And besides, prime-factoring accomplishes the same thing in a much shorter way. But somehow I think my conception is qualitatively different.

    I also consider a "straight line" to be the perimeter of a circle whose radius is infinity.

    I must be out of my mind.

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