# Conjecture for prime pairs of difference two

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

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
Homework Helper
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.

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.

That would be the said "Twin Primes Conjecture".

HallsofIvy