Skip to main content
Log in

Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers II

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

Abstract

In this paper, we find two integers k 0, m of 159 decimal digits such that if kk 0 (mod m), then none of five consecutive odd numbers k, k −2, k − 4, k − 6 and k − 8 can be expressed in the form 2n ± p α, where p is a prime and n, α are nonnegative integers.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Romanoff, N. P.: Über einige Sätze der additiven Zahlentheorie. Math. Ann., 57, 668–678 (1934)

    Article  MathSciNet  Google Scholar 

  2. Erdős, P.: On integers of the form 2r + p and some related problems. Summa Brasil. Math., 2, 113–123 (1950)

    MathSciNet  Google Scholar 

  3. Cohen, F., Selfridge, J. L.: Not every number is the sum or difference of two prime powers. Math. Comput., 29, 79–81 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, Y. G.: Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers. Math. Comput. 74, 1025–1031 (2005)

    MATH  Google Scholar 

  5. Chen, Y. G.: On integers of the form \( 2^n \pm p_1^{\alpha _1 } \cdots p_r^{\alpha _r } \). Proc. Amer. Math. Soc., 128, 1613–1616 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, Y. G.: On integers of the form k2n + 1. Proc. Amer. Math. Soc., 129, 355–361 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, Y. G.: On integers of the forms k − 2n and k2n + 1. J. Number Theory, 89, 121–125 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, Y. G.: On integers of the forms k r − 2n and k r2n + 1. J. Number Theory, 98, 310–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, Y. G.: On integers of the forms k ± 2n and k2n ± 1. J. Number Theory, 125, 14–25 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Y. G., Sun, X. G.: On Romanoff’s constant. J. Number Theory, 106, 275–284 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Erdős, P., Odlyzko, A. M.: On the density of odd integers of the form (p − 1)2n and related questions. J. Number Theory, 11, 257–263 (1979)

    Article  MathSciNet  Google Scholar 

  12. Granville, A., Soundararajan, K.: A binary additive problem of Erdős and the order of 2 (mod p 2). The Ramanujan J., 2, 283–298 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Guy, R. K.: Unsolved problems in number theory, 2nd ed. Springer, New York, 1994

    MATH  Google Scholar 

  14. Jaeschke, G.: On the smallest k such that all k·2N + 1 are composite. Math. Comput., 40, 381–384 (1983); corrigendum, Math. Comput., 45, 637 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Stanton, R. G., Williams, H. C.: Further results on covering of the integer 1 + k2n by primes, in Combinatorial Math. VIII, Lecture Notes in Math. 884, Springer-Verlag, Berlin/New York, 1980, 107–114

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yong Gao Chen.

Additional information

Supported by the National Natural Science Foundation of China, Grant No 10471064 and 10771103

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Y.G., Tang, M. Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers II. Acta. Math. Sin.-English Ser. 24, 1883–1890 (2008). https://doi.org/10.1007/s10114-008-6456-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6456-1

Keywords

MR(2000) Subject Classification

Navigation