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Smooth numbers with few nonzero binary digits

Part of: Multiplicative number theory Exponential sums and character sums Elementary number theory

Published online by Cambridge University Press:  20 June 2023

Maximilian Hauck  and
Igor E. Shparlinski Open the ORCID record for Igor E. Shparlinski [Opens in a new window]
Maximilian Hauck
Affiliation:
Department of Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany e-mail: max.hauck01@gmail.com
Igor E. Shparlinski*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: igor.shparlinski@unsw.edu.au
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Abstract

We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.

Keywords

Smooth integerssparse binary representationscharacter sums

MSC classification

Primary: 11A63: Radix representation; digital problems 11L40: Estimates on character sums 11N25: Distribution of integers with specified multiplicative constraints
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 74 - 89
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work started during a very enjoyable visit by the authors to the Max Planck Institute for Mathematics, Bonn, whose hospitality and support is very much appreciated. During the preparation of this work, I.S. was also supported in part by the Australian Research Council Grant DP200100355.

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