Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 439–455

Density of solutions to quadratic congruences



A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k > 1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree nx with k prime factors such that a fixed quadratic equation has exactly 2k solutions modulo n.


Dirichlet’s theorem asymptotic density primes in arithmetic progression squarefree number 

MSC 2010

11D45 11B25 11N37 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008.MATHGoogle Scholar
  2. [2]
    H. Kornblum, E. Landau: Über die Primfunktionen in einer arithmetischen Progression. Math. Zeitschr. 5 (1919), 100–111. (In German.)CrossRefMATHGoogle Scholar
  3. [3]
    E. Landau: Sur quelques problèmes relatifs à la distribution des nombres premiers. S. M. F. Bull. 28 (1900), 25–38. (In French.)MATHGoogle Scholar
  4. [4]
    H. L. Montgomery, R. C. Vaughan: Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.Google Scholar
  5. [5]
    C. Pomerance: On the distribution of amicable numbers. J. Reine Angew. Math. 293/294 (1977), 217–222.MathSciNetMATHGoogle Scholar
  6. [6]
    P. Ribenboim: The New Book of Prime Number Records, Springer, New York, 1996.CrossRefMATHGoogle Scholar
  7. [7]
    E. M. Wright: A simple proof of a theorem of Landau. Proc. Edinb. Math. Soc., II. Ser. 9 (1954), 87–90.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Indian Institute of Science Education and ResearchPuneIndia

Personalised recommendations