Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is a non-negative integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression

and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo d containing a's coprime to d.

Dirichlet's theorem does not require that the sequence contains only prime numbers and deals with infinite sequences. For finite sequences, there exist arbitrarily long arithmetic progressions of primes, a theorem known as the Green–Tao theorem.

An integer is a prime for the Gaussian integers if either the square of its modulus is a prime number (in the normal sense) or one of its parts is zero and the absolute value of the other is a prime that is congruent to 3 modulo 4. The primes (in the normal sense) of the type 4n + 3 are (sequence in the OEIS)

They correspond to the following values of n: (sequence in the OEIS)

...
Wikipedia