Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T09:04:05.591Z Has data issue: false hasContentIssue false
  • English
  • Français

EIGHT CONSECUTIVE POSITIVE ODD NUMBERS NONE OF WHICH CAN BE EXPRESSED AS A SUM OF TWO PRIME POWERS

Part of: Sequences and sets

Published online by Cambridge University Press:  08 June 2009

YONG-GAO CHEN
YONG-GAO CHEN*
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove the following result: there exists an infinite arithmetic progression of positive odd numbers such that for any term k of the sequence and any nonnegative integer n, each of the 16 integers k−2n, k−2−2n, k−4−2n, k−6−2n, k−8−2n, k−10−2n, k−12−2n, k−14−2n, k2n−1, (k−2)2n−1, (k−4)2n−1, (k−6)2n−1, (k−8)2n−1, (k−10)2n−1, (k−12)2n−1 and (k−14)2n−1 has at least two distinct odd prime factors; in particular, for each term k, none of the eight integers k, k−2, k−4, k−6, k−8, k−10, k−12 or k−14 can be expressed as a sum of two prime powers.

Keywords

covering systemsodd numberssums of prime powers

MSC classification

Secondary: 11B25: Arithmetic progressions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 237 - 243
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

This work was supported by the National Natural Science Foundation of China, grant no. 10771103.

References

[1] Chen, Y.-G., ‘On integers of the forms k±2n and k2n±1’, J. Number Theory 125 (2007), 14–25.CrossRefGoogle Scholar
[2] Chen, Y.-G., ‘Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers’, Math. Comp. 74 (2005), 1025–1031.CrossRefGoogle Scholar
[3] Chen, Y.-G. and Tang, M., ‘Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers II’, Acta Math. Sin. (Engl. Ser.) 24 (2008), 1883–1890.CrossRefGoogle Scholar
[4] Cohen, F. and Selfridge, J. L., ‘Not every number is the sum or difference of two prime powers’, Math. Comp. 29 (1975), 79–81.CrossRefGoogle Scholar
[5] Erdős, P., ‘On integers of the form 2r+p and some related problems’, Summa Brasil. Math. 2 (1950), 113–123.Google Scholar
[6] Filaseta, M., Finch, C. and Kozek, M., ‘On powers associated with Sierpinski numbers, Riesel numbers and Polignacs conjecture’, J. Number Theory 128 (2008), 1916–1940.CrossRefGoogle Scholar
[7] Guy, R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).CrossRefGoogle Scholar
[8] Luca, F. and Stănică, P., ‘Fibonacci numbers that are not sums of two prime powers’, Proc. Amer. Math. Soc. 133 (2005), 1887–1890.CrossRefGoogle Scholar
[9] Romanoff, N. P., ‘Über einige Sätze der additiven Zahlentheorie’, Math. Ann. 109 (1934), 668–678.CrossRefGoogle Scholar
[10] Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. 3 (1892), 265–284.CrossRefGoogle Scholar