Dirichlet’s Theorem for arithmetic progressions with difference 4

From antiquity, Euclid’s Theorem asserts that the number of primes is infinite. Since all primes except are odd, a re-phrasement of Euclid’s theorem is that the set of odd integers contains an infinite number of primes. A natural generalization, proved by Dirichlet, is that every arithmetic progression of the form contains an infinite number of primes, provided that and are relatively prime. (Dirichlet’s Theorem) If , then the arithmetic progression contains infinitely many primes, where denotes the greatest common divisor of and . The proof is one of the first application of Fourier analysis on finite abelian groups. Here in this post, we prove Dirichlet’s theorem when , i.e., both the arithmetic […]