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](https://en.wikipedia.org/wiki/Twin_prime).)


Is $f$ computable?



Computability of prime difference function
I once heard Harvey Friedman suggest that the set of prime-differences, that is, the set of all natural numbers $n$ for which there are primes $p,q$ with $p-q=n$, as a possible candidate for all we knew for an intermediate Turing degree — a noncomputable set between $0$ and $0'$ — that was *natural*, not specifically constructed to have that feature. 


