Conjecture for prime pairs of difference two

  • #1
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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?
 

Answers and Replies

  • #2
CRGreathouse
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This is the of-yet-unproved Twin Prime Conjecture.
 
  • #3
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primes separated by only one even number

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.
 
  • #4
CRGreathouse
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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.

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.
 
  • #5
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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.

The number of pairs of primes with a difference of two.
 
  • #6
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That would be the said "Twin Primes Conjecture".
 
  • #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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