Advertisement

Frontiers of Mathematics in China

, Volume 14, Issue 5, pp 1017–1035 | Cite as

Slim exceptional set for sums of two squares, two cubes, and two biquadrates of primes

  • Rui ZhangEmail author
Research Article
  • 11 Downloads

Abstract

We prove that, with at most \(O(N^{\frac{17}{192}+\varepsilon})\) exceptions, all even positive integers up to N are expressible in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , where p1, p2, …, p6 are prime numbers. This gives large improvement of a recent result \(O(N^{\frac{13}{16}+\varepsilon})\) due to M. Zhang and J. J. Li.

Keywords

Waring-Goldbach problem circle method exceptional set 

MSC

11P05 11P32 11P55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to express the most sincere gratitude to Prof. Zhixin Liu for his valuable advice and constant encouragement. This work was supported by the National Natural Science Foundation of China (Grant No. 11871367).

References

  1. 1.
    Hua L K. Some results in the additive prime number theory. Quart J Math, 1938, 9: 68–80MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hua L K. Additive Theory of Prime Numbers. Providence: Amer Math Soc, 1965zbMATHGoogle Scholar
  3. 3.
    Kawada K, Wooley T D. Relations between exceptional sets for additive problems. J Lond Math Soc, 2010, 82: 437–458MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kumchev A V. On Weyl sums over primes and almost primes. Michigan Math J, 2006, 54: 243–268MathSciNetCrossRefGoogle Scholar
  5. 5.
    Li T Y. Enlarged major arcs in the Waring-Goldbach problem. Int J Number Theory, 2016, 12: 205–217MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu J Y. Enlarged major arcs in additive problems. Math Notes, 2010, 88: 395–401MathSciNetCrossRefGoogle Scholar
  7. 7.
    Liu Y H. On a Waring-Goldbach problem involving squares, cubes and biquadrates. Bull Korean Math Soc, 2018, 55: 1659–1666MathSciNetzbMATHGoogle Scholar
  8. 8.
    Liu Y H. Exceptional set for sums of unlike powers of primes. Int J Number Theory, 2019, 15(2): 339–352MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu Z X. Goldbach-Linnik type problems with unequal powers of primes. J Number Theory, 2017, 176: 439–448MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lü X D. Waring-Goldbach problem: two squares, two cubes and two biquadrates. Chinese Ann Math Ser A, 2015, 36: 161–174 (in Chinese)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ren X M. On exponential sums over primes and application in Waring-Goldbach problem. Sci China Ser A, 2005, 48: 785–797MathSciNetCrossRefGoogle Scholar
  12. 12.
    Vaughan R C. On the representation of numbers as sums of powers of natural numbers. Proc Lond Math Soc, 1970, 21: 160–180MathSciNetCrossRefGoogle Scholar
  13. 13.
    Vaughan R C. The Hardy-Littlewood Method. 2nd ed. Cambridge: Cambridge Univ Press, 1997CrossRefGoogle Scholar
  14. 14.
    Vinogradov I M. Elements of Number Theory. New York: Dover Publications, 1954zbMATHGoogle Scholar
  15. 15.
    Zhang M, Li J J. Exceptional set for sums of unlike powers of primes. Taiwanese J Math, 2018, 22: 779–811MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhao L L. The additive problem with one cube and three cubes of primes. Michigan Math J, 2014, 63: 763–779MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhao L L. On the Waring-Goldbach problem for fourth and sixth powers. Proc Lond Math Soc, 2014, 108: 1593–1622MathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinChina

Personalised recommendations