Dirichlet's theorem on primes in arithmetic progressions

In terms of arithmetic progressions
Any arithmetic progression of positive integers, where the terms of the arithmetic progression are relatively prime to the common difference, contains infinitely many prime numbers.

In terms of congruence
Given two relatively prime positive integers $$a$$ and $$m$$, there exist infinitely many primes $$p$$ such that $$p \equiv a \mod m$$.

The case that is easiest to prove, and also the most useful, is the case where $$a = 1$$, i.e., given any $$m$$, there are infinitely many primes congruent to 1 mod $$m$$.