TITLE: Is $k+p$ prime infinitely many times?
QUESTION [10 upvotes]: I have the following conjecture:

Let $k\in\mathbb{N}$ be even. Now $k+p$ is prime for infinitely many primes $p$. 

I couldn't find anything on this topic, but I'm sure this has been thought of before. I tried to solve this using Dirichlet's theorem on arithmetic progressions and the Green–Tao theorem, but no luck with those. Is this question equivalent to an existing open problem? If not, how can I prove this (I prefer hints, but I appreciate full answers, too)?
 - Edit -
As has been pointed out in the comments, this is not a duplicate. I'm asking for infinitely many primes $p$ such that $p+k$ is prime, not only one.

REPLY [2 votes]: There are some results known. It has been proven that there is at least one $k \leq 246$ that appears infinitely often as a prime gap. 
Furthermore, assuming respectively the Elliott–Halberstam conjecture and its generalisation, one can prove that there is at respectively at least one $k \leq 12$ and $k \leq 6$ that appears infinitely often as a prime gap.
Of course, a prime gap is stronger than your condition, since the primes don't have to be consecutive.