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Mathematicians Team Up To Close the Prime Gap 194

Posted by Unknown Lamer
from the there-can-be-only-4680 dept.
Hugh Pickens DOT Com writes "On May 13, an obscure mathematician garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers. Yitang Zhang showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers, representing a major leap toward proving the centuries-old twin primes conjecture, which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13). Now Erica Klarreich reports at Quanta Magazine that other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower. By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million. Then Terence Tao, a winner of the Fields Medal, mathematics' highest honor, created a 'Polymath project,' an open, online collaboration to improve the bound that attracted dozens of participants. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower. Zhang's work and, to a lesser degree, Maynard's fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn't be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record. 'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."
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Mathematicians Team Up To Close the Prime Gap

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  • by Anonymous Coward on Wednesday November 20, 2013 @09:55AM (#45472025)

    We cannot allow a prime gap!

  • by Anonymous Coward on Wednesday November 20, 2013 @09:57AM (#45472037)

    sometimes its better to go it alone, then come back to the group with your results so that someone else may profit from them.

    sometimes its better to be a part a group in order to establish your ideas and discuss, then go it alone when the group holds you back.

  • Nice work (Score:5, Funny)

    by Anonymous Coward on Wednesday November 20, 2013 @10:05AM (#45472079)

    If they keep this shit up, pretty soon they will prove that every number is prime.

  • See Kuhn (Score:4, Informative)

    by TheloniousToady (3343045) on Wednesday November 20, 2013 @10:05AM (#45472085)

    'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."

    History is rife with examples of the lone genius making a leap forward, thereby allowing the crowd to take it even further. See The Structure of Scientific Revolutions [wikipedia.org] by Thomas Kuhn.

    • by Anonymous Coward

      Favourite quote re said crowd taking it further:

      I am not a Kuhnian.
      --Thomas Kuhn

    • Re:See Kuhn (Score:5, Informative)

      by sfkaplan (1004665) on Wednesday November 20, 2013 @11:55AM (#45473067) Homepage

      Wait, what? If that's what you think Kuhn wrote, then you may need to go re-read the book.

      His central claim was not that lone geniuses make leaps, but that leaps can rarely be attributed so clearly to a single individual, moment, or event. The Big Idea of that book is that the process of scientific advance is much messier, and much more contextually dependent, then we are lead to believe in popular accounts. Often the so-called "genius moments" are a critical step, but not easily or correctly identified as such until after the fact, making it hard to know *which* insight was really the critical one.

      There's lots of dispute about Kuhn, but let's not make matters worse by incorrectly describing what he wrote.

      • by yusing (216625)

        We should all go re-read Kuhn anyways. Because (thinking of the current state of cosmology for one) clearly we didn't get it yet.

      • Actually, I haven't read it yet. I only recently heard about it while reading a biography of Joseph Priestley [wikipedia.org]. I hope to read the Kuhn book soon.

        Anyway, sorry if I got that wrong. I was just trying to further the always-erudite discussion here. To that end, thanks for setting me straight in the most condescending and pedantic way possible. ;-)

  • Yawn. (Score:2, Funny)

    by Anne_Nonymous (313852)

    >> which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13)

    Yawn. Call me when you find a set of primes separated by one.

    • by xxxJonBoyxxx (565205) on Wednesday November 20, 2013 @10:13AM (#45472155)

      Er...2 and 3. What do I win?

      • by MiniMike (234881) on Wednesday November 20, 2013 @10:28AM (#45472281)

        What do I win?

        The tattered remnants of Anne_Nonymous's (probably not her real name) Geek Card, in a frame.

      • A "misses the concept of infinity" patch to sew on your uniform.

        But we all know the final lower bound will be 42 anyway.

    • by beelsebob (529313)

      Okay... Found them. 2 and 3.

    • {2, 3}

    • by ChristW (18232)

      2, 3

      Thank you, I'll be here all week...

    • I've found a set of primes separated by one.

      {2,3}

      Do I get a Fields Medal for that?

      Also, mathematics is awesome. Even if I can't understand it.

    • Re:Yawn. (Score:4, Funny)

      by Anonymous Coward on Wednesday November 20, 2013 @10:24AM (#45472263)

      I can do better: I can prove that there are infinitely many pairs of prime numbers p and q separated by zero!

      Here are the first few such pairs:

      (2,2)
      (3,3)
      (5,5)
      (7,7)

  • Factoring Primes (Score:2, Interesting)

    by ebno-10db (1459097)

    Will they ever learn to factor prime numbers though? I understand it's difficult, but solving it would save a lot of embarrassment when people misstate the problem.

    • by mrego (912393)
      Prime numbers are the scaffolding of the number system and all of nature. I am convinced that twin primes represent the spine or backbone of an understanding of prime numbers. If the mystery of twin primes fall, soon we will understand primes and then all of nature. And we'll be rich, evil, mad geniuses with unlimited power. Mhhhahahahaha
      • we'll be rich, evil, mad geniuses with unlimited power. Mhhhahahahaha

        Sounds good, as long as we don't actually achieve it. The joy of being an evil genius is in striving, not succeeding.

      • The 'spine or backbone' of prime numbers is most certainly tied not to the twin primes, but the quad primes.

        I submit, that there are an infinite number of quad prime sets.

        [ P,P+2,P+6,P+8 that are all prime, aka, back to back pairs of twin primes]

    • Re:Factoring Primes (Score:5, Informative)

      by Anonymous Coward on Wednesday November 20, 2013 @10:35AM (#45472319)

      Factoring prime numbers is dead easy. Here's an implementation in Python:

      # factorize prime
      # precondition: the argument is a prime
      # if the precondition is not met, the result is wrong.
      # result: The factorization of the argument.
      def factorPrime(p):
          return [ p ];

      It's the factoring of composite numbers that is difficult.

      Actually, even factorizing composite numbers isn't really difficult. It's just difficult to do it in a way that finishes before you stop caring about the result. ;-)

      • It's the factoring of composite numbers that is difficult.

        Actually it's even easier

        def factornumber(n):
                return [ n,1 ];

        (And now I can't want to see how someone out pedantics-me in continuing this petty-up-man-ship thread.)

        • And now I can't want to see how someone out pedantics-me in continuing this petty-up-man-ship thread.

          Done before you posted - see upthread. Just as there's an Obfuscated C contest, Slashdot should have an "Ultimate Pedantry" contest.

          • by Valdrax (32670)

            That was done before you posted; see up-thread. Just as there's an International Obfuscated C Code Contest, Slashdot should have an "Ultimate Pedantry" contest.

            FTFY.

    • ``The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers.'' -Bill Gates, The Road Ahead, pg. 265

      Perhaps, he was educated as to the stupidity of his remark later.

  • Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, (...) to push the bound even lower.

    600 ought to be enough for anyone.

  • by ebno-10db (1459097) on Wednesday November 20, 2013 @10:14AM (#45472169)

    Three people are asked to prove that all of the odd numbers are prime - a physicist, a mathematician and a programmer.

    The physicist goes first. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a ... oops, experimental error, 11 is a prime ...".

    Next the mathematician takes a crack at it: "3 is a prime, 5 is a prime, 7 is a prime, and the rest by induction".

    Finally it's the programmer's turn. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime ...".

    • by VortexCortex (1117377) <VortexCortexNO@S ... t-retrograde.com> on Wednesday November 20, 2013 @10:39AM (#45472337)

      // [VC 2013.11.20] Fix primary oddity error in prime oddity test.
      #define 9 015

      • by ebno-10db (1459097) on Wednesday November 20, 2013 @11:47AM (#45472983)

        An interesting paradox. You're not a real programmer if you realized that define was necessary, but you are a real programmer if you obfuscated it using that archaic octal notation.

        • by Luyseyal (3154)

          Archaic? I ran into that bug^H^H^Hnotation in JavaScript, of all things. Must be NEWWWWWWWW.

          -l

          /man that was an annoying bug to fix

          • by EETech1 (1179269)

            My first assembly program for the Atmel AVR gave me quite the headache debugging! The output would jump around randomly, and sometimes go backwards.

            My program worked fine if I put the data tables in hex, but if I tried to put them in (what I thought was) decimal all hell broke loose. Watching the code in the emulator, I couldn't figure out why the lookups were giving me the wrong numbers.

            I finally just made two data tables from 0 - 255, one in hex and one in decimal, and then programmed, and read back the

    • by jalopezp (2622345)
      It's not a programmer, it's an engineer. A programmer what is proof.
      • It's not a programmer, it's an engineer.

        You're right, I should have stuck to the original version. With the character as an engineer, it's a humorous error. With a programmer, it's more like what did you expect?

    • Close, but the programmer would have likely introduced a spelling error.

  • Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

    So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.

    • by Anonymous Coward

      Zhang proved it's finite. The others have just lowering the finite number with newer proofs.

    • Didn't you see the phrase "lesser extent"?
    • At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

      It was Yitang Zhang who settled the original long-standing open question - that being, is there any number such that you will always find pairs of primes separated by that number or less. The ultimate goal is to solve the twin prime conjecture - bringing the number in question down to 2.

      Your own wording is a little confusing - I'm not sure who the "eight" are, or whether "eight - James Maynard" refers to seven mathematicians, in which you couldn't describe them all as "an obscure mathematician" ;)

      His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers

      This (from

      • (such as 70,000,000! and it's neighbour)

        Err, ignore this. Getting confused.

    • Re:Summary (Score:5, Insightful)

      by Kjella (173770) on Wednesday November 20, 2013 @11:13AM (#45472657) Homepage

      Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.

      No. Before May 2013 there was no proof on an infinite pair of primes being a finite bound apart.
      - May 2013: Zhang, bound 70 million
      - End of May 2013: Others, bound <60 million
      - July 2013: Terence Tao & Polymath project: bound 4680
      - Now: James Maynard, bound 600
      - Twin conjencture: still unproven, bound 2

      So the "big" discovery was Zhang, for managing to put a bound on it in the first place. The rest are improvements on that proof, but not very fundamental ones. Proving the twin conjencture would be huge, but nobody's done that yet. The Polymath project and probably many others are working on it. The conjencture is almost certainly true, but notoriously hard to prove. Probably the easiest "feel" to get for it is the Sieve of Eratosthenes, make a long list of odd numbers then strike out the multiples of primes. Once you strike out the 3s it'll be obvious you don't get triplets since 3, 9, 15, 21, 27 and so on are all multiples of 3 so the "candidates" are (5,7) (11,13), (17,19), (23,25) and so on. As you add more primes like 25 = 5*5 it'll get fewer and fewer pairs but they keep occuring rather randomly. It feels like that with infinite primes they'll randomly end up being next to each other an infinite number of times, but proving it is another matter. For example if you take the Fibonacci sequence (1,1,2,3,5,8,13,21...) it's obvious it's an infinite series but the distance between numbers also grows to infinity. Not so with primes, by these proofs.

    • Re:Summary (Score:5, Insightful)

      by gnasher719 (869701) on Wednesday November 20, 2013 @11:13AM (#45472665)

      So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.

      Here's what it real means: There were conjectures, one of them famous, which stated:

      There are infinitely many pairs (p, p+2) of consecutive primes.
      There are infinitely many pairs (p, p+4) of consecutive primes.
      There are infinitely many pairs (p, p+6) of consecutive primes.
      ...
      There are infinitely many pairs (p, p+600) of consecutive primes.

      It is now proven that at least one of these conjectures is true.

    • by readin (838620)
      Explanation by car analogy:

      You're driving on a highway leaving a city. At every prime numbered mile marker there's a gas station. As you leave the city the gas stations are close together, with a station at the 2 mile marker, another at the 3 mile marker, another at the 5 mile marker, etc. As you get into the suburbs the gas stations are less frequent. As you get into the desert you find that gas stations are hard to find.

      But you notice something - it seems that no matter how far you drive into the de
  • by ctrl-alt-canc (977108) on Wednesday November 20, 2013 @10:47AM (#45472403)
    They are not so sexy [wikipedia.org], after all...
    • Only a mathematician would think that two primes separated by six is "sexy". And they say programmers and engineers are a sad lot.

  • If N is between 2 and 10^260, then the number of primes less than N is more than N / 600. So in that range the _average_ gap between consecutive primes is less than 600. For N = 10^20 it is actually quite rare that the gap between two consecutive primes is over 600.
  • by museumpeace (735109) on Wednesday November 20, 2013 @12:20PM (#45473273) Journal
    so little of what news is dragged before me these days does much to make me hopeful of humanity's prospects on this planet. This story is the rare exception. We could be a great species. We could solve what looked for centuries to be impossible problems. We could...

    Thanks /. This story was not in any of my regular channels today.
    • True, stories like this, and those related to my crack-smoking mayor, give me a reason to get up each morning.

  • Good news, everybody! It's been now proven, that there's an infinite number of one-hundred-times sexier primes [wikipedia.org], so there's enough for all of you lonely geeks :)

    You've got to find them, though...

  • by AdamHaun (43173) on Wednesday November 20, 2013 @04:35PM (#45475713) Journal

    The linked abstracts are pretty vague. Are there any mathematicians here who can explain how (seemingly arbitrary) large numbers like 600 or 70 million come out of these proofs? People are saying they're all tweaks of the same basic method, so what is that basic method, exactly?

  • Excepting 33 small numbers, all even numbers can be represented as the sum of two odd primes, BOTH of which are members of a twin prime set. The 33 exceptions of course have solutions to the normal Goldbach Conjecture, it is just that one or both of the primes do not have a corresponding twin.

    Note if you can find the proof of this, then you have killed multiple birds with one stone.

    You get the infiinite twins problem solved.
    You get the Goldback conjecture solved.
    And you find that is also shows that th

  • ...does this affect encryption in some way? My understanding is that a lot of encryption relies on the difficulty of finding prime numbers. I may be wrong. (It's certainly not my specialty.)
  • Why isn't cold fjord in here blaming Snowden for all this?

You can do this in a number of ways. IBM chose to do all of them. Why do you find that funny? -- D. Taylor, Computer Science 350

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