Skip to main content
SpringerLink
Log in

On sums of a prime and four prime squares in short intervals

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we prove that each sufficiently large integer N ≢ 1 (mod 3) can be written as

$$ N = p + p_1^2 + p_2^2 + p_3^2 + p_4^2 , $$

, with

$$ \left| {p - \frac{N} {5}} \right| \leqslant U,\left| {p_j - \sqrt {\frac{N} {5}} } \right| \leqslant U,j = 1,2,3,4, $$

where U = N 9/20 + ɛ and p, p j are primes.

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. Hua, L. K.: Some results in the additive prime number theory. Quart. J. Math. (Oxford), 9, 68–80 (1938)

    Article  Google Scholar 

  2. Meng, X. M.: On sums of a prime and four prime squares in short intervals. Acta Math. Hungar., 105, 261–283 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Liu, J. Y., Zhan, T.: On sums of five almost equal prime squares. Acta Arith., 77, 369–383 (1996)

    MathSciNet  Google Scholar 

  4. Liu, J. Y., Zhan, T.: Hua’s theorem on prime squares in short intervals. Acta Mathematica Sinica, English Series, 16, 669–690 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Liu, J. Y., Zhan, T.: Sums of five almost equal prime squares. II, Sci. China Ser. A, 41, 710–722 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu, J. Y., Lü, G. S., Zhan, T.: Exponential sums over primes in short intervals. Sci. China Ser. A, 49, 449–457 (2006)

    Google Scholar 

  7. Bauer, C.: Sums of five almost equal prime squares. Acta Mathematica Sinica, English Series, 21, 833–840 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ren X. M., Tsang, K. M.: Waring-Goldbach problem for unlike powers. Acta Mathematica Sinica, English Series, 23, 265–280 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lü, G. S.: Hua’s Theorem with five almost equal prime variables. Chinese Ann. Math. Ser. B, 26, 291–304 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lü, G. S.: Hua’s Theorem on five almost equal prime squares. Acta Mathematica Sinica, English Series, 22, 907–916 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lü, G. S.: Sums of nine almost equal prime cubes. Acta Mathematica Sinica, Chinese Series, 49, 195–204 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Lü, G. S.: Estimation of exponential sums over primes in short intervals. Acta Mathematica Sinica, Chinese Series, 49, 693–698 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Lü, G. S., Lao, H. X.: On exponential sums over primes in short intervals. Monatsh. Math., 151, 153–164 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lü, G. S., Xu, Y. F.: Hua’s theorem with nine almost equal prime variables. Acta Math. Hungar., 116, 309–326 (2007)

    Article  MathSciNet  Google Scholar 

  15. Liu, J. Y., Wooley, T. D., Yu, G.: The quadratic Waring-Goldbach problem. J. Number Theory, 107, 298–321 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, J. Y.: On Lagrange’s theorem with prime variables, Quart. J. Math. (Oxford), 54, 454–462 (2003)

    Article  Google Scholar 

  17. Liu, J. Y., Zhan, T.: The exceptional set in Hua’s theorem for three squares of primes. Acta Mathematica Sinica, English Series, 21, 335–350 (2005)

    Article  MathSciNet  Google Scholar 

  18. Wooley, T. D.: Slim exceptional sets for sums of four squares. Proc. London Math. Soc., 85, 1–21 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gallagher, P. X.: A large sieve density estimate near σ = 1. Invent. Math., 11, 329–339 (1970)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shandong University, Ji’nan, 250100, P. R. China

    Guang Shi Lü

  2. Department of Statistics and Mathematics, Shandong Finance Institute, Ji’nan, 250014, P. R. China

    Xian Meng Meng

Authors
  1. Guang Shi Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xian Meng Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guang Shi Lü.

Additional information

This work is supported by the National Natural Science Foundation of China (Grant No. 10701048)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lü, G.S., Meng, X.M. On sums of a prime and four prime squares in short intervals. Acta. Math. Sin.-English Ser. 24, 1291–1302 (2008). https://doi.org/10.1007/s10114-008-6089-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6089-4

Keywords

MR(2000) Subject Classification

Search

Navigation