<abstract xmlns="http://www.w3.org/1999/xhtml">

Let <italic>s</italic>(<italic>n</italic>) be the number of nonzero bits in the binary digital expansion of the integer <italic>n</italic>. We study, for fixed <italic>k, ℓ, m</italic>, the Diophantine system
<italic>s</italic>(<italic>ab</italic>)= <italic>k, s</italic>(<italic>a</italic>)= <italic>ℓ,</italic> and <italic>s</italic>(<italic>b</italic>)= <italic>m</italic>
in odd integer variables <italic>a, b</italic>.When <italic>k</italic> =2 or <italic>k</italic> = 3, we establish a bound on <italic>ab</italic> in terms of <italic>ℓ</italic> and <italic>m</italic>. While such a bound does not exist in the case of <italic>k</italic> =4, we give an upper bound for min{<italic>a, b</italic>} in terms of <italic>ℓ</italic> and <italic>m</italic>.
</abstract>

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system
s(ab)= k, s(a)= ℓ, and s(b)= m
in odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of ℓ and m.

Products of Integers with Few Nonzero Digits

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Products of Integers with Few Nonzero Digits"}

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system
s(ab)= k, s(a)= ℓ,...

Products of Integers with Few Nonzero Digits

Published Online: May 31, 2022

Page range: 11 - 28

Received: Jun 30, 2021

Accepted: Nov 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0006

Keywords
sum of digits, digital expansion, factors

© 2022 Hajime Kaneko et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Products of Integers with Few Nonzero Digits

Published Online: May 31, 2022

Page range: 11 - 28

Received: Jun 30, 2021

Accepted: Nov 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0006

Keywordssum of digits, digital expansion, factors

© 2022 Hajime Kaneko et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
sum of digits, digital expansion, factors