## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 35, Number 1 (2006), 249-259.

### Primes in Certain Arithmetic Progressions

#### Abstract

We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.

#### Article information

**Source**

Funct. Approx. Comment. Math. Volume 35, Number 1 (2006), 249-259.

**Dates**

First available in Project Euclid: 16 December 2008

**Permanent link to this document**

http://projecteuclid.org/euclid.facm/1229442627

**Digital Object Identifier**

doi:10.7169/facm/1229442627

**Mathematical Reviews number (MathSciNet)**

MR2271617

**Zentralblatt MATH identifier**

05784932

**Subjects**

Primary: 11A41: Primes

Secondary: 11C08: Polynomials [See also 13F20] 11R04: Algebraic numbers; rings of algebraic integers 11R18: Cyclotomic extensions

**Keywords**

Dirichlet's theorem prime divisors of polynomials Chebotarev density theorem

#### Citation

Murty, Ram; Thain, Nithum. Primes in Certain Arithmetic Progressions. Funct. Approx. Comment. Math. 35 (2006), no. 1, 249--259. doi:10.7169/facm/1229442627. http://projecteuclid.org/euclid.facm/1229442627.