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Conjecture for prime pairs of difference two

  1. Jun 2, 2009 #1
    Can it be proven that the number of prime pairs with a difference of two (that is, primes separated by only one even number) approaches infinity?
     
  2. jcsd
  3. Jun 2, 2009 #2

    CRGreathouse

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    This is the of-yet-unproved Twin Prime Conjecture.
     
  4. Jun 3, 2009 #3
    Do you mean that infinitely many primes separated by the same even number? Or do you mean infinitely many primes separated by a multiple of the same even number? The latter is true.
     
  5. Jun 3, 2009 #4

    CRGreathouse

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    I know Elliott-Halberstam implies that (via Goldston-Pintz-Yıldırım), but is it known unconditionally? As far as I know, [itex]g_n>\sqrt{\log p_n}[/itex] for all n sufficiently large has not been disproven.

    Oh wait, I just reread what you wrote. The latter is trivially true, since all prime gaps but the first are divisible by 2.
     
  6. Jun 4, 2009 #5
    The number of pairs of primes with a difference of two.
     
  7. Jun 4, 2009 #6
    That would be the said "Twin Primes Conjecture".
     
  8. Jun 4, 2009 #7

    HallsofIvy

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    But you wouldn't say "approaches" infinity in either case. Either the number of twin primes is infinite or it is a specific integer.
     
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