Dirichlet's theorem on primes in arithmetic progressions
From Number
Contents
- 1 Statement
- 2 Related facts
- 2.1 Stronger facts
- 2.2 Easy cases
- 2.3 Related facts about infinitude
- 2.4 Conjectures/facts about the first Dirichlet prime
- 2.5 Conjectures/facts about Bertrand's postulate on Dirichlet primes
- 2.6 Conjectures/facts about contiguous blocks of Dirichlet primes
- 2.7 Conjectures that generalize to higher degree polynomials
Statement
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
- There are infinitely many primes that are one modulo any modulus
- Dirichlet's theorem for modulus four
- Dirichlet's theorem for modulus eight
Related facts about infinitude
Conjectures/facts about the first Dirichlet prime
- Heath-Brown's conjecture on the first Dirichlet prime: A conjecture, saying that the first Dirichlet prime in a given congruence class modulo
is
.
- Chowla's conjecture on the first Dirichlet prime: A conjecture, saying that the first Dirichlet prime in a given congruence class modulo
is
.
- Chowla's corollary to generalized Riemannn hypothesis: Proved conditional to the generalized Riemann hypothesis, saying that the first Dirichlet prime in a given congruence class modulo
is
.
- Linnik's theorem: An unconditional theorem, saying that there exists
such that the first Dirichlet prime in a given congruence class modulo
is
. Heath-Brown showed that we can take
.
Conjectures/facts about Bertrand's postulate on Dirichlet primes
Conjectures/facts about contiguous blocks of Dirichlet primes
- Green-Tao theorem: This states that for any
, there exists a prime arithmetic progression of length
: an arithmetic progression of length
, all of whose members are 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.