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?

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    $\begingroup$ Do you know the answer for just one sequence $\lfloor\alpha_1 n+\beta_1\rfloor$? $\endgroup$
    – Wojowu
    Commented Aug 2, 2021 at 18:33
  • $\begingroup$ @Wojowu Yes, this is known. I believe it follows from bounds for exponential sums over primes, once you express the condition of being in the Beatty sequence in terms of additive characters. // If $\alpha_1/\alpha_2$ is a rational $a/b$, then if $p_1 = \lfloor \alpha_1 n + \beta_1\rfloor$ and $p_2 =\lfloor \alpha_2 n + \beta_n\rfloor $ then $b p_2 -a p_1$ takes only finitely many different values. For each of them, proving there are infinitely many $p_1,p_2$ satisfying that equation should be basically equivalent in difficulty to twin primes. $\endgroup$
    – Will Sawin
    Commented Aug 2, 2021 at 18:43
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    $\begingroup$ @Wojowu For just one sequence, you can take a look at this paper by Banks and Shparlinski: arxiv.org/pdf/0708.1015.pdf, specifically Corollary 5.3 $\endgroup$ Commented Aug 2, 2021 at 19:12


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