Functiones et Approximatio Commentarii Mathematici

Primes in Certain Arithmetic Progressions

Ram Murty and Nithum Thain

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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

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

First available in Project Euclid: 16 December 2008

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Mathematical Reviews number (MathSciNet)

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Digital Object Identifier

Primary: 11A41: Primes
Secondary: 11C08: Polynomials [See also 13F20] 11R04: Algebraic numbers; rings of algebraic integers 11R18: Cyclotomic extensions

Dirichlet's theorem prime divisors of polynomials Chebotarev density theorem


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.

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