Products of integers with few nonzero digits

TL;DR

This paper investigates the solutions to a Diophantine system involving the binary digit sum of integers and their products, establishing bounds for certain cases and highlighting differences when the sum exceeds three.

Contribution

It provides bounds on the product of integers with prescribed binary digit sums for specific cases, advancing understanding of binary digit sum Diophantine systems.

Findings

Bound on $ab$ when $k=2$ or $k=3$ in terms of $ ext{l}$ and $m$.

No such bound exists for $k=4$, but an upper bound for $ ext{min}\{a,b ext{}$ is given.

The results differentiate between cases with small and larger binary digit sums.

Abstract

Let $s(n)$ be the number of nonzero bits in the binary digital expansion of the integer $n$. We study, for fixed $k,\ell,m$, the Diophantine system $$ s(ab)=k, \quad s(a)=\ell,\quad \mbox{and }\quad s(b)=m, $$ in odd integer variables $a,b$. When $k=2$ or $k=3$, we establish a bound on $ab$ in terms of $\ell$ 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 $\ell$ and $m$.