Dirichlet's theorem

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:

{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,...}

Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by Gauss and proved by Dirichlet in 1835 with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory.

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

See also:

• Linnik's theorem (1944)