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.

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.
