Czechoslovak Mathematical Journal, online first, 17 pp.

Density of solutions to quadratic congruences

Neha Prabhu

Received December 31, 2015.   First published May 5, 2017.

Neha Prabhu, Indian Institute of Science Education and Research, Dr Homi Bhabha Rd, NCL Colony, Pashan, Pune, Maharashtra 411008, India, e-mail:

Abstract: 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 $n\leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.

Keywords: Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number

Classification (MSC 2010): 11D45, 11B25, 11N37

DOI: 10.21136/CMJ.2017.0712-15

Full text available as PDF.

  [1] G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2008). MR 2445243 | Zbl 1159.11001
  [2] H. Kornblum, E. Landau: Über die Primfunktionen in einer arithmetischen Progression. Math. Zeitschr. 5 (1919), 100-111. (In German.) DOI 10.1007/BF01203156 | MR 1544375 | Zbl 47.0154.02
  [3] E. Landau: Sur quelques problèmes relatifs à la distribution des nombres premiers. S. M. F. Bull. 28 (1900), 25-38. (In French.) MR 1504359 | Zbl 31.0200.01
  [4] H. L. Montgomery, R. C. Vaughan: Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge (2007). DOI 10.1017/CBO9780511618314 | MR 2378655 | Zbl 1142.11001
  [5] C. Pomerance: On the distribution of amicable numbers. J. Reine Angew. Math. 293/294 (1977), 217-222. DOI 10.1515/crll.1977.293-294.217 | MR 0447087 | Zbl 0349.10004
  [6] P. Ribenboim: The New Book of Prime Number Records. Springer, New York (1996). DOI 10.1007/978-1-4612-0759-7 | MR 1377060 | Zbl 0856.11001
  [7] E. M. Wright: A simple proof of a theorem of Landau. Proc. Edinb. Math. Soc., II. Ser. 9 (1954), 87-90. DOI 10.1017/S0013091500021349 | MR 0065579 | Zbl 0057.28601

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