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Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are prime numbers? How about a special case when $b=d=1$?

This seems like something that should hold, some kind of a generalization of Dirichlet's theorem, but I wasn't able to show it.

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    $\begingroup$ It's an open problem. If you could do the case $a=c=1$, $b$ any large integer and $d=b+2$, you'd solve the twin prime conjecture. $\endgroup$ – Felipe Voloch Sep 7 '15 at 16:51
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    $\begingroup$ This isn't always true. Taking $a=b=c=1,d=2$ we get $k+3,k+4$ and for $k\geq 0$ they can't both be prime. $\endgroup$ – Wojowu Sep 7 '15 at 17:05
  • $\begingroup$ You may find this interesting: arxiv.org/abs/1509.01564 $\endgroup$ – Sylvain JULIEN Sep 7 '15 at 19:09
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This is a special case of Dickson's conjecture (for suitable $a,b,c,d$, cf. Wojowu's comment below your post), and as Felipe Voloch remarked, it is an open problem. There have been some striking recent advances towards this conjecture, check out the work Green-Tao and Maynard.

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