Murty, M. Ram; Thain, Nithum Prime numbers in certain arithmetic progressions. (English) Zbl 1196.11011 Funct. Approximatio, Comment. Math. 35, 249-259 (2006). From the text: The authors 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). The first author [J. Madras Univ., Sect. B 51, 161–169 (1988; Zbl 1194.11093)] 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 (as it is difficult to obtain), they consider generalizations of this question to algebraic number fields. The main tool will be the Chebotarev density theorem showing that the class of polynomials needed for a Euclidean proof does not exist unless \(\ell^2\equiv 1\pmod k\). Cited in 3 Documents MSC: 11A41 Primes 11C08 Polynomials in number theory 11R04 Algebraic numbers; rings of algebraic integers 11R18 Cyclotomic extensions Keywords:Dirichlet’s theorem; prime divisors of polynomials; Chebotarev density theorem Citations:Zbl 1194.11093 PDFBibTeX XMLCite \textit{M. R. Murty} and \textit{N. Thain}, Funct. Approximatio, Comment. Math. 35, 249--259 (2006; Zbl 1196.11011) Full Text: DOI Euclid References: [1] M. Bauer, Zur Theorie der algebraischen Zahlkoerper , Math. Annalen, 77 (1916), 353–356. · JFM 46.0249.02 · doi:10.1007/BF01475865 [2] P. Bateman and M.E. Low, Prime numbers in arithmetic progression with difference 24 , Amer. Math. Monthly, 72 (1965), 139–143. JSTOR: · Zbl 0127.26805 · doi:10.2307/2310975 [3] K. Conrad, Euclidean Proofs of Dirichlet’s Theorem , Website:\hb http://www.math.uconn.edu/ kconrad/blurbs/dirichleteuclid.pdf. · Zbl 1133.14028 [4] I. Gerst and J. Brillhart, On the prime divisors of polynomials , Amer. Math. Monthly, 78 (1971), 250–266. JSTOR: · Zbl 0214.30604 · doi:10.2307/2317521 [5] G.H. Hardy & E.M. Wright, An introduction to the theory of numbers , 4th Edition, Oxford, 1960. · Zbl 0086.25803 [6] M.R. Murty, On the existence of “Euclidean proofs” of Dirichlet’s theorem on primes in arithmetic progressions , B. Sc. Thesis, 1976, (unpublished) Carleton University. [7] M.R. Murty, Primes in Certain Arithmetic Progressions , J. Madras Univ., Section B, 51 (1988), 161–169. · Zbl 1194.11093 [8] M.R. Murty & J. Esmonde, Problems in Algebraic Number Theory , Second Edition, Graduate Texts in Mathematics 190 , (2005). · Zbl 1055.11001 [9] T. Nagell, Sur les diviseurs premiers des polynômes , Acta Arith, 15 (1969), 235–244. · Zbl 0181.05501 [10] I. Schur, Uber die Existenz unendlich vieler Primzahlen in einiger speziellen arithmetischen Progressionen , S-B Berlin. Math. Ges., 11 (1912), 40–50. · JFM 43.0261.02 [11] B. Wyman, What is a reciprocity law , Amer. Math. Monthly, 79 (1972), 571–586. JSTOR: · Zbl 0244.12010 · doi:10.2307/2317083 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.