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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: ygchen@njnu.edu.cn)
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Abstract

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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

Information

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

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