Dirichlet's theorem in number theory states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + n d, where n > 0, or in other words: there are infinitely many primes which are congruent a mod d.
This theorem extends Euclid's theorem that there are infinitely many primes (in this case of the form 3 + 4n, which are also the Gaussian primes, or of the form 1 + 2n, for every odd numbers, excluding 1). Note that the theorem does not say that there are infinitely many consecutive terms in the arithmetic progression a, a+d, a+2d, a+3d, ..., which are prime. For example in the Euclid's proof with Gaussian primes we get primes 'just' for n from:
In algebraic number theory Dirichlet's theorem generalizes to Chebotarev's density theorem.