Conjecture for prime pairs of difference two

1. Jun 2, 2009

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?

2. Jun 2, 2009

CRGreathouse

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

3. Jun 3, 2009

Dragonfall

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. Jun 3, 2009

CRGreathouse

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.

5. Jun 4, 2009

Loren Booda

The number of pairs of primes with a difference of two.

6. Jun 4, 2009

Dragonfall

That would be the said "Twin Primes Conjecture".

7. Jun 4, 2009

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.