Abstract
B. J. Birch [1] proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p a q b, where p, q are given coprime integers greater than 1. Subsequently, Davenport pointed out that the exponent b can be bounded in terms of p and q. N. Hegyvári [3] gave an effective version of this bound. In this paper, we improve this bound by reducing one step.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. J. Birch, Note on a problem of Erdős, Proc. Cambridge Philos. Soc., 55 (1959), 370–373.
J. H. Fang, A note on the completeness of an exponential type sequence, Chin. Ann. Math. Ser. B, 32 (2011), 527–532.
N. Hegyvári, On the completeness of an exponential type sequence, Acta Math. Hungar., 86 (2000), 127–135.
V. H. Vu, Some new results on subset sums, J. Number Theory, 124 (2007), 229–233.
Author information
Authors and Affiliations
Corresponding author
Additional information
Corresponding author.
The authors are supported by the National Natural Science Foundation of China (Grant No. 11071121) and the second author is also supported by Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 11KJB110006).
Rights and permissions
About this article
Cite this article
Chen, YG., Fang, JH. Remark on the completeness of an exponential type sequence. Acta Math Hung 136, 189–195 (2012). https://doi.org/10.1007/s10474-011-0188-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-011-0188-x