Skip to main content
SpringerLink
Log in

The exceptional set for integers of the form \([p_1^c]+[p_2^c]\)

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Let \(1< c < 24/19\). We show that the number of integers \(n \le N\) that cannot be written as \([p_1^c] + [p_2^c]\) (\(p_1\), \(p_2\) primes) is \(O(N^{1-\sigma +\varepsilon })\). Here \(\sigma \) is a positive function of c (given explicitly) and \(\varepsilon \) is an arbitrary positive number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.C. Baker, Sums of two relatively prime cubes. Acta Arith. 129, 103–149 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  2. R.C. Baker, Some Diophantine equations and inequalities with primes. Funct. Approx. Comment. Math. 64, 203–250 (2021)

    Article  MathSciNet  Google Scholar 

  3. Y.C. Cai, A ternary Diophantine inequality involving primes. Int. J. Number Theory 14, 2257–2268 (2018)

    Article  MathSciNet  Google Scholar 

  4. S.W. Graham, G. Kolesnik, Van der Corput’s Method of Exponential Sums (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  5. D.R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34, 1365–1377 (1982)

    Article  MathSciNet  Google Scholar 

  6. D.R. Heath-Brown, A new \(k\)-th derivative estimate for a trigonometric sum via Vinogradov’s integral. Proc. Steklov Inst. Math. 296, 88–103 (2017)

    Article  MathSciNet  Google Scholar 

  7. M.B.S. Laporta, On a binary problem with prime numbers. Math. Balkanica (N.S.) 13, 119–123 (1999)

    MathSciNet  Google Scholar 

  8. W. B. Zhu, Representation of integers as sums of fractional powers of primes and powers of 2, Acta Arith.181, 185–196 (2017), erratum, ibid., 197–199

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Roger Baker

Authors
  1. Roger Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger Baker.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baker, R. The exceptional set for integers of the form \([p_1^c]+[p_2^c]\). Period Math Hung 88, 127–136 (2024). https://doi.org/10.1007/s10998-023-00543-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-023-00543-4

Keywords

Mathematics Subject Classification

Search

Navigation