A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided there are no congruence conditions which immediately preclude this. I'm curious if anyone has investigated the analogous conjecture for Beatty sequences, i.e.

**Conjecture.** Let $\alpha_1,\beta_1,\alpha_2,\beta_2\in\mathbb{R}$ with $\alpha_1,\alpha_2$ irrational. Then there are infinitely many $n$ for which $\lfloor\alpha_1 n+\beta_1\rfloor$ and $\lfloor\alpha_2 n+\beta_2\rfloor$ are both prime.

It seems likely that this is comparable in difficulty to the twin prime conjecture, but perhaps there are reasons why it might be more tractable. Does anyone know of work that has been done on this problem?

onesequence $\lfloor\alpha_1 n+\beta_1\rfloor$? $\endgroup$