Advertisement

Primes of the form \({kM^n+n}\)

  • X.-G. SunEmail author
Article
  • 14 Downloads

Abstract

Erdős and Odlyzko proved that odd integers k such that \({k2^n +1}\) is prime for some positive integer n have a positive lower density. We prove that for sufficiently large x, the number of integers k\({\leq}\)x such that k is relatively prime to M and such that \({kM^n+n}\) is prime for some positive integer n is at least C(M)x for some constant C(M) depending only on M.

Keywords and phrases

asymptotic density prime Selberg’s sieve method 

Mathematics Subject Classification

11A07 11B25 11P32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Banks, W., Finch, C., Luca, F., Pomerance, C., Stănică, P.: Sierpiński and Carmichael numbers. Trans. Amer. Math. Soc. 367, 355–376 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Y.G.: On integers of the form \(k2^{n}+1\). Proc. Amer. Math. Soc. 129, 355–361 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Y.G.: On integers of the forms \(k-2^{n}\) and \(k2^{n}+1\). J. Number Theory 89, 121–125 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y.G.: On integers of the forms \(k^{r}-2^{n}\) and \(k^{r}2^{n}+1\). J. Number Theory 98, 310–319 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y.G.: On integers of the forms \(k\pm 2^{n}\) and \(k2^{n}\pm 1\). J. Number Theory 125, 14–25 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cilleruelo, J., Luca, F., Pizarro-Madariaga, A.: Carmichael numbers in the sequence \(\{2^{n}k + 1\}_{n>1}\). Math. Comp. 85, 357–377 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős, P., Odlyzko, A.M.: On the density of odd integers of the form \((p-1)2^{-n}\) and related questions. J. Number Theory 11, 257–263 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer (New York, 2004)Google Scholar
  9. 9.
    H. Halberstam and H. E. Richert, Sieve Methods, London Math. Soc. Monographs, No. 4, Academic Press (London–New York, 1974)Google Scholar
  10. 10.
    Luca, F., Stănică, P.: On numbers of the form \(p+2^{n}-n\). J. Comb. Number Theory 6, 157–162 (2016)zbMATHGoogle Scholar
  11. 11.
    W. Sierpiński, Sur un problème concernant les nombres \(k2^{n}+1\), Elem. Math., 15 (1960), 73–74; Corrigendum, 17 (1962), 85.Google Scholar
  12. 12.
    Sun, X.G., Fang, J.H.: On the density of integers of the form \((p-1)2^{-n}\) in arithmetic progressions. Bull. Austral. Math. Soc. 78, 431–436 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.School of SciencesHuaiHai Institute of TechnologyJiangsuPeople’s Republic of China

Personalised recommendations