Dirichlet's theorem on arithmetic progressions

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Dirichlet's theorem states that for two positive integers a and q ≥ 1, which are coprime, there are infinitely many primes of the form a + n q, for an nonnegative integer n = 0, 1, 2, 3, ..., or in other words there are infinitely many primes, which are congruent a mod q.

The theorem was first conjectured by Gauss and proved by Dirichlet in 1835 with Dirichlet's L-Series. Today the proof remains almost in its original form and it is still considered to be difficult to understand. The theorem represents the beginning of an analytic number theory.

This theorem does not say there are infinitely many consecutive terms in this arithmetic progression a, a+q, a+2q, a+3q, ..., which are primes. Chowla proved this for the case of three consecutive terms.