Functiones et Approximatio Commentarii Mathematici

Primes in Certain Arithmetic Progressions

Ram Murty and Nithum Thain

Full-text: Open access

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: 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.


Export citation

References

  • M. Bauer, Zur Theorie der algebraischen Zahlkoerper, Math. Annalen, 77 (1916), 353--356.
  • P. Bateman and M.E. Low, Prime numbers in arithmetic progression with difference 24, Amer. Math. Monthly, 72 (1965), 139--143.
  • K. Conrad, Euclidean Proofs of Dirichlet's Theorem, Website:\hb http://www.math.uconn.edu/~kconrad/blurbs/dirichleteuclid.pdf.
  • I. Gerst and J. Brillhart, On the prime divisors of polynomials, Amer. Math. Monthly, 78 (1971), 250--266.
  • G.H. Hardy & E.M. Wright, An introduction to the theory of numbers, 4th Edition, Oxford, 1960.
  • 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 Thesis, 1976, (unpublished) Carleton University.
  • M.R. Murty, Primes in Certain Arithmetic Progressions, J. Madras Univ., Section B, 51 (1988), 161--169.
  • M.R. Murty & J. Esmonde, Problems in Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics 190, (2005).
  • T. Nagell, Sur les diviseurs premiers des polynômes, Acta Arith, 15 (1969), 235--244.
  • I. Schur, Uber die Existenz unendlich vieler Primzahlen in einiger speziellen arithmetischen Progressionen, S-B Berlin. Math. Ges., 11 (1912), 40--50.
  • B. Wyman, What is a reciprocity law, Amer. Math. Monthly, 79 (1972), 571--586.