Dirichlet's theorem on primes in arithmetic progressions

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Template:Infinitude fact


Let be relatively prime natural numbers. Then, there exist infinitely many primes such that:


For fixed , the primes that are congruent to modulo are termed Dirichlet primes.

Related facts

Stronger facts

Easy cases

Related facts about infinitude

Conjectures/facts about the first Dirichlet prime

Conjectures/facts about Bertrand's postulate on Dirichlet primes

Conjectures/facts about contiguous blocks of Dirichlet primes

Conjectures that generalize to higher degree polynomials

  • Bunyakovsky conjecture is a conjecture for polynomials of degree two or more whose analogue for linear polynomials would be Dirichlet's theorem.