Conjecture for prime pairs of difference two

[Loren Booda]

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?

[CRGreathouse]

This is the of-yet-unproved Twin Prime Conjecture.

[Dragonfall]

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.

[CRGreathouse]

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, g_n>\sqrt{\log p_n} 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.

[Loren Booda]

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.

[Dragonfall]

That would be the said "Twin Primes Conjecture".

[HallsofIvy]

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