**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 + 4*n*, which are also the Gaussian primes, or of the form 1 + 2*n*, 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*+2*d*, *a*+3*d*, ..., which are prime. For example in the Euclid's proof with Gaussian primes we get primes 'just' for *n* from:

- {1,2,4,5,7,10,11,14,16,17,19,20,25,26,31,32,34,37,40,41,44,47,49,52,55,56,59,62,65,67,70,76,77,82,86,89,91,94,95,...}

In algebraic number theory Dirichlet's theorem generalizes to Chebotarev's density theorem.

**See also:**