Functiones et Approximatio Commentarii Mathematici

Primes in Certain Arithmetic Progressions

Ram Murty and Nithum Thain

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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

Mathematical Reviews number (MathSciNet)
MR2271617

Zentralblatt MATH identifier
05784932

Digital Object Identifier
doi:10.7169/facm/1229442627

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. Functiones et Approximatio Commentarii Mathematici 35 (2006), no. 1, 249--259. doi:10.7169/facm/1229442627. http://projecteuclid.org/euclid.facm/1229442627.


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