Products of Integers with Few Nonzero Digits

Kaneko, Hajime; Stoll, Thomas

doi:10.2478/udt-2022-0006

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.

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Products of Integers with Few Nonzero Digits

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