Translational equilibria between primes and composites.

Question

Think of the primes as bin $P$ and the composites as bin $C$, and let $0 \in P$. We can already say that given $\sigma : \Bbb{Z} \to \Bbb{Z} : x \mapsto x + 2$

$P^C = \{ p \in P : \sigma(p) \in C \} \simeq C^P = \{ c \in C : \sigma(c) \in P\}$ can't we?

Or is this also hard like the twin prime conjecture?

Answer 1

There are infinitely many primes $p \equiv 1 \pmod 3$. For each such prime $p$, $p+2$ is a multiple of $3$. It follows that $P^C$ is infinite.

There are infinitely many primes $p \equiv 2 \pmod 3$. For each such prime $p$, $p-2$ is a multiple of $3$. It follows that $C^P$ is infinite.