Can there ever exist infinite sets of integers $A, B$ such that $A + B = \{ a + b: a \in A, b \in B\} = \Bbb{P}$? Where $\Bbb{P}$ is the set of odd primes? You can include $0$ and / or $\pm$ odd primes if you must. Maybe you can include $2$, I'm still not sure.. Confused!
This seems maybe easy, or maybe very difficult. I will work on my answer and update this here.
New notation, let $\Bbb{P}_o = $ the odd primes.
Here's working up finitely, an example.
$$A + B = \{2, 4, 8, 14, \dots\} \\ + \\ \{3, 5, 9, 15, \dots \} = \{ p \in \Bbb{P}: p \geq 5\} = \Bbb{P}_o \setminus \{3\} $$
The pattern seems to be ?
No. If $A$ and $B$ each have numbers with two residues $\bmod 3$ there will be a sum that is a multiple of $3$ and not prime, so at least one is all the same residue class. The other must have two residues so we get the primes that are $1 \bmod 3$ and $2 \bmod 3$. In neither case can we get $3$ as a sum.