Possible values of difference of 2 primes Is it true that for any even number $2k$, there exists primes $p, q$ such that $p-q = 2k$?
Polignac's conjecture talks about having infinitely many consecutive primes whose difference is $2k$. This has not been proven or disproven.
This is a more general version of my question on the possible value of prime gaps.

Of course, the odd case is easily done.
 A: In short, this is an open problem (makes Polignac seem really tough then, doesn't it?).
The sequence A02483 of the OEIS tracks this in terms of $a(n) =$ the least $p$ such that $p + 2n = q$. On the page it mentions that this is merely conjectured.
In terms of a positive result, I am almost certain that Chen's theorem stating that every even number can be written as either $p + q$ or $p + q_1q_2$ is attained by sieving methods both powerful enough and loose enough to also give that every even number can be written as either a difference of two primes or a difference of a prime and an almost-prime (or of an almost-prime and a prime). I think I've even seen this derived before.
This would come as a corollary of Polignac's conjecture or of the vastly stronger Schinzel's hypothesis H, which is one of those conjectures that feels really far from reach to me. I suppose that it has been proved on average over function fields (I think), so perhaps that's hopeful. On the other hand, so has the Riemann Hypothesis.
