Dirichlet's Theorem
Theorem
For any positive integers and such that , there exists infinitely many prime such that
Hence, for any arithmetic progression, unless it obviously contains finitely many primes (first term and common difference not coprime), it contains infinitely many primes.
Stronger Result
For any positive integers and such that , where the sum is over all primes less than that are congruent to mod , and is the totient function.