# Dirichlet's theorem on primes in arithmetic progressions

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## 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.