Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 713–724 | Cite as

On Romanov’s constant

  • Christian Elsholtz
  • Jan-Christoph Schlage-Puchta


We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly one representation of the form \(p+2^{k}\) has positive density. Previous results of this kind needed “at most 15” in place of “exactly one”. To achieve these results we introduce a new method. In particular we make use of uneven distribution of sums of a power of two and a reduced residue class.

Mathematics Subject Classification

11P32 Goldbach-type theorems Other additive questions involving primes 


  1. 1.
    Chen, Y.G., Sun, X.G.: On Romanoff’s constant. J. Number Theory 106, 275–284 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    de Polignac, A.: Recherches nouvelles sur les nombres premiers. C. R. Acad. Sci. Paris 29, 397–401, 738–739 (1849)Google Scholar
  3. 3.
    Erdős, P.: On integers of the form \(2^k + p\) and some related problems. Summa Brasil. Math. 2, 113–125 (1950)MathSciNetGoogle Scholar
  4. 4.
    Erdős, P., Turán, P.: Ein zahlentheoretischer Satz. Mitt. Forsch. Inst. Math. Mech. Univ. Tomsk 1, 101–103 (1935)MATHGoogle Scholar
  5. 5.
  6. 6.
    Habsieger, L., Roblot, X.-F.: On integers of the form \(p+2^k\). Acta Arith. 122(1), 45–50 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Habsieger, L., Sivak-Fischler, J.: An effective version of the Bombieri–Vinogradov theorem, and applications to Chen’s theorem and to sums of primes and powers of two. Arch. Math. 95(6), 557–566 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Halberstam, H., Richert, H.-E.: Sieve Methods. London Mathematical Society Monographs. Academic Press, London (1974)MATHGoogle Scholar
  9. 9.
    Khalfalah, A., Pintz, J.: On the representation of Goldbach numbers by a bounded number of powers of two. Elementare und analytische Zahlentheorie, 129–142. Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag, Stuttgart (2006)Google Scholar
  10. 10.
    Klimov, N.I.: Upper estimates of some number theoretical functions. Dokl. Akad. Nauk SSSR (N.S.) 111, 16–18 (1956). (Russian)MathSciNetMATHGoogle Scholar
  11. 11.
    Lü, G.S.: On Romanoff’s constant and its generalized problem. Adv. Math. (Beijing) 36, 94–100 (2007)MathSciNetGoogle Scholar
  12. 12.
    Pintz, J.: A note on Romanov’s constant. Acta Math. Hung. 112, 1–14 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Romani, F.: Computations concerning primes and powers of two. Calcolo 20, 319–336 (1984)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Romanoff, N.P.: Über einige Sätze der additiven Zahlentheorie. Math. Ann. 109, 668–678 (1934)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sun, X.G.: On the density of integers of the form \(2^k+p\) in arithmetic progressions. Acta Math. Sin. 26(1), 155–160 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Van Der Corput, J.G.: On de Polignac’s conjecture. Simon Stevin 27, 99–105 (1950)MathSciNetGoogle Scholar
  17. 17.
    Wu, J.: Chen’s double sieve, Goldbach’s conjecture and the twin prime problem. Acta Arith. 114, 215–273 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Christian Elsholtz
    • 1
  • Jan-Christoph Schlage-Puchta
    • 2
  1. 1.Institut für Mathematik ATechnische Universität GrazGrazAustria
  2. 2.Mathematical InstituteUniversity RostockRostockGermany

Personalised recommendations