Consider the following function $f: \omega\to \{0,1\}$:

Set $f(n) = 1$ if for all $k\in \omega$ there are prime numbers $p,q > k$ such that $n = p-q$; and set $f(n) = 0$ otherwise.

(Trivially, if $n$ is odd, we have $f(n) = 0$. Moreover, the question whether $f(2) = 1$ is the subject of the twin prime conjecture.)

Is $f$ computable?