Inspired by this question and this answer I raise the following conjecture:
If $\{f_1,\dots,f_n\}$ is a set of entire holomorphic functions and if for every $m\in\mathbb N$, it exists a function with $f_i(m)\in\mathbb P$, then some function $f_k$ is a constant.
Can this be proved?
This is false. Set $f(x)=2\cos^2(\pi x)$. This is an entire function that is prime at every natural number.