Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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


Author: Yong-Gao Chen
Journal: Math. Comp. 74 (2005), 1025-1031
MSC (2000): Primary 11A07, 11B25
Published electronically: July 20, 2004
MathSciNet review: 2114663
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term $M$ of which none of five consecutive odd numbers $M, M-2, M-4, M-6$ and $M-8$ can be expressed in the form $2^n \pm p^\alpha $, where $p$ is a prime and $n, \alpha $ are nonnegative integers.


References [Enhancements On Off] (What's this?)

  • 1. A. S. Bang, Taltheoretiske Unders$\phi $gelser, Tidsskrift for Mat. (5), 4(1886), 70-80, 130-137.
  • 2. G. D. Birkhoff and H. S. Vandiver, On the integral divisors of $a^{n}-b^{n}$, Ann. Math. 5(1904), 173-180.
  • 3. Y. G. Chen, On integers of the form $2^{n} \pm p_{1}^{\alpha _{1} } \cdots p_{r}^{\alpha _{r} }$, Proc. Amer. Math. Soc. 128(2000), 1613-1616. MR 2000j:11006
  • 4. Y. G. Chen, On integers of the form $k2^{n}+1$, Proc. Amer. Math. Soc. 129(2001), 355-361. MR 2003a:11004
  • 5. Y. G. Chen, On integers of the forms $k-2^{n}$ and $k2^{n}+1$, J. Number Theory 89(2001), 121-125. MR 2002b:11020
  • 6. Y. G. Chen, On integers of the forms $k^{r}-2^{n}$and $k^{r}2^{n}+1$, J. Number Theory 98(2003), 310-319. MR 2003m:11004
  • 7. F. Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29(1975), 79-81. MR 51:12758
  • 8. P. Erdos, On integers of the form $2^{r} + p$ and some related problems, Summa Brasil. Math. 2(1950), 113-123. MR 13:437i
  • 9. R. K. Guy, ``Unsolved problems in number theory," 2nd ed., Springer, New York, 1994. MR 96e:11002
  • 10. G. Jaeschke, On the smallest $k$ such that all $k\cdot 2^{N}+1$ are composite, Math. Comput. 40(1983), 381-384; corrigendum, Math. Comput. 45(1985) 637.MR 84k:10006; MR 87b:11009
  • 11. N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57(1934), 668-678.
  • 12. R. G. Stanton and H. C. Williams, Further results on covering of the integer $1+k2^{n}$ by primes, in ``Combinatorial Math. VIII," Lecture Notes in Math. 884, Springer-Verlag, Berlin/New York, 1980, 107-114. MR 84j:10009
  • 13. K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. 3(1892), 265-284.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11A07, 11B25

Retrieve articles in all journals with MSC (2000): 11A07, 11B25


Additional Information

Yong-Gao Chen
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, Peoples Republic of China
Email: ygchen@pine.njnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-04-01674-6
Keywords: Covering systems, odd numbers, sums of prime powers
Received by editor(s): January 2, 2003
Received by editor(s) in revised form: October 2, 2003
Published electronically: July 20, 2004
Additional Notes: Supported by the National Natural Science Foundation of China, Grant No. 10171046 and the Teaching and Research Award Program for Outstanding Young Teachers in Nanjing Normal University
Article copyright: © Copyright 2004 American Mathematical Society