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.



Primes in simultaneous arithmetic progressions
This is a special case of [Dickson's conjecture](https://en.wikipedia.org/wiki/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](http://arxiv.org/abs/math/0606088) and [Maynard](http://arxiv.org/abs/1405.2593v2).


