Dirichlet's Theorem On Arithmetic Progressions

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 ≥ 0. In other words, there are infinitely many primes which 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 arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges, and that different arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among each congruence class modulo d.

Note that Dirichlet's theorem does not require the prime numbers in an arithmetic sequence to be consecutive. It is also known that there exist arbitrarily long finite arithmetic progressions consisting only of primes, but this is a different result, known as the Green–Tao theorem.