Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 150–163 | Cite as

Sums of four prime cubes in short intervals

  • A. LanguascoEmail author
  • A. Zaccagnini


We prove that a suitable asymptotic formula for the average number of representations of integers \(n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}\), where \(p_1\), \(p_2\), \(p_3\), \(p_4\) are prime numbers, holds in intervals shorter than the the ones previously known.

Key words and phrases

Waring–Goldbach problem Hardy–Littlewood method 

Mathematics Subject Classification

primary 11P32 secondary 11P55 11P05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Brüdern, A sieve approach to the Waring-Goldbach problem. I. Sums of four cubes, Ann. Sci. École Norm. Sup. (4), 28 (1995), 461–476Google Scholar
  2. 2.
    Davenport, H.: On Waring's problem for cubes. Acta Math. 71, 123–143 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gambini, A., Languasco, A., Zaccagnini, A.: A Diophantine approximation problem with two primes and one \(k\)-th power of a prime. J. Number Theory 188, 210–228 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hua, L.K.: Some results in the additive prime number theory. Quart. J. Math. Oxford 9, 68–80 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    L. K. Hua, Additive Theory of Prime Numbers, Trans. Math. Monographs, vol. 13, Amer. Math. Soc. (Providence, RI, 1965)Google Scholar
  6. 6.
    A. Languasco and A. Zaccagnini, Short intervals asymptotic formulae for binary problems with primes and powers. II: density \(1\), Monatsh. Math., 181 (2016), 419–435Google Scholar
  7. 7.
    Languasco, A., Zaccagnini, A.: Sum of one prime and two squares of primes in short intervals. J. Number Theory 159, 1945–1960 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, H., Zhao, F.: Density of integers that are the sum of four cubes of primes in short intervals. Acta Math. Hungar. 151, 8–23 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, Z.: Density of the sums of four cubes of primes. J. Number Theory 132, 735–747 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Amer. Math. Soc. (Providence, RI, 1994)Google Scholar
  11. 11.
    Ren, X.: Sums of four cubes of primes. J. Number Theory 98, 156–171 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. reine angew. Math. 591, 1–20 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    K. F. Roth, On Waring's problem for cubes, Proc. London Math. Soc. (2), 53 (1951), 268–279Google Scholar
  14. 14.
    Vaughan, R.C.: The Hardy-Littlewood Method, 2nd edn. Cambridge Univ, Press (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Tullio Levi-Civita”Universitá di PadovaPadovaItaly
  2. 2.Dipartimento di Scienze Matematiche, Fisiche e InformaticheUniversitá di ParmaParmaItaly

Personalised recommendations