Dirichlet's theorem on arithmetic progressions tells us that there are infinitely many primes in an arithmetic progression $a+nd$ for coprime positive integers $a,d\in\mathbb{N}$ fixed and $n\in\mathbb{N}_0$ varying. Generalisations of the theorem even tells us how those primes are distributed. But I have yet to see anything on coupled arithmetic progressions. Take for example \begin{align} 13 + 6n \\ 7 + 3n \\ 5 + 2n \end{align} We know that each arithmetic progression has infinitely many primes - but how about the number of simultaneous primes, i.e. for how many values of $n$ are all three progressions prime? From my computations, it seems there are a lot of simultaneous primes: We have $819$ for $13+6n<200000$. Below I will list the first $25$ for anyone interested. My conjecture is that there are infinitely many.
In general, for $k$ non-trivial arithmetic progressions \begin{align*} a_1 &+ nd_1 \\ a_2 &+ nd_2 \\ &. . . \\ a_k &+ nd_k \end{align*} what can we say about the number of primes? In which cases could it be infinite? It is certainly sometimes infinite, such as for $k=1$.
Note that answering this question completely is very hard. Indeed, stating that \begin{align*} 2 + 5n \\ 4 + 5n \end{align*} has infinitely many simultaneous primes, just as an example, would be a positive proof of a generalisation of the twin prime conjecture. Requiring that all the $d_i$'s are different seems reasonable. This way, we do not have to solve the twin prime conjecture, and we avoid trivial counterexamples like \begin{align} 1+5n\\ 2+5n \end{align} which clearly have no simultaneous primes. Off the top of my head, I could not come up with a set of coupled arithmetic progressions with distinct $d_i$'s such that the number of simultaneous primes is finite.
I will also gladly take any references about this topic, as I have not been able to find any.
List of simultaneous primes for the first set of coupled arithmetic progressions: $(13, 7, 5), (37, 19, 13), (85, 43, 29), (121, 61, 41), (157, 79, 53), (217, 109, 73), (301, 151, 101), (325, 163, 109), (445, 223, 149), (541, 271, 181), (697, 349, 233), (841, 421, 281), (877, 439, 293), (1045, 523, 349), (1201, 601, 401), (1225, 613, 409), (1261, 631, 421), (1345, 673, 449), (1381, 691, 461), (1621, 811, 541), (1705, 853, 569), (1837, 919, 613), (1981, 991, 661), (2017, 1009, 673), (2101, 1051, 701).$