Are there any distinct $a, b$ s.t. $a + x$ prime $\Longleftrightarrow$ $b + x$ prime?

Question

Are there any distinct $a, b \in \mathbb{N}$ s.t. $a + x$ prime $\Longleftrightarrow$ $b + x$ prime for all $x \in \mathbb{N_0}$?

I can show there are no coprime $a,b$ using Dirichlet's theorem: Choose $x = ib$. Then $b + ib$ is never prime, but there are infinitely many primes of the form $a + ib$.

Answer 1

Assume such $a,b$ exist, wlog. $d:=b-a>0$. Let $p$ be a prime $>a$. Then $a+(p-a)$ prime implies $b+(p-a)=p+d$ is prime. Hence by induction $p+nd$ is primes for all $n\in\Bbb N_0$. In particular $p+pd=p(d+1)$ is prime, which is absurd.