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 


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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

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