I know that Dirichlet's theorem says that there are infinitely many primes $p$ such that $p\equiv a$ (mod $n$) when gcd$(a,n)=1$. I'm wondering more generally about a system of power congruences
$x^{k_1}\equiv a_1$ (mod $n_1$)
$x^{k_2}\equiv a_2$ (mod $n_2$)
:
$x^{k_m}\equiv a_m$ (mod $n_m$)
Given fixed positive integers $k_i,a_i,n_i$ as above, is there a condition, e.g., $gcd(a_i,lcm(n_1,\ldots,n_m))=1$, that guarantees the existence of infinitely many primes $p$ such that $p^{k_i}\equiv a_i$ (mod $n_i$) for all $1\leq i\leq m$?
Maybe this should be a separate post, but I'm also wondering about conditions that guarantee that the above system has any solutions $x\in\{0,1,\ldots,L-1$}, prime or not, where $L=lcm(n_1,\ldots,n_m)$.