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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The binary digits of a power


Author: Kenneth B. Stolarsky
Journal: Proc. Amer. Math. Soc. 71 (1978), 1-5
MSC: Primary 10A40
MathSciNet review: 495823
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Abstract: Let $ B(m)$ denote the number of ones in the binary expansion of the integer $ m \geqslant 1$ and let $ {r_h}(m) = B({m^h})/B(m)$ where h is a positive integer. The maximal order of magnitude of $ {r_h}(m)$ is $ c(h){(\log m)^{(h - 1)/h}}$ where $ c(h) > 0$ depends only on h. That this is best possible follows from the Bose-Chowla theorem. The minimal order of magnitude of $ {r_2}(m)$ is at most $ c{(\log \log m)^2}/\log m$ where $ c > 0$ is an absolute constant.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0495823-5
PII: S 0002-9939(1978)0495823-5
Article copyright: © Copyright 1978 American Mathematical Society