Generalizations
The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-order polynomials. Whether or not even simple quadratic polynomials such as x2 + 1 attain infinitely many prime values is an important open problem.
In algebraic number theory, Dirichlet's theorem generalizes to Chebotarev's density theorem.
Read more about this topic: Dirichlet's Theorem On Arithmetic Progressions