Smooth numbers with few non-zero binary digits

TL;DR

This paper demonstrates that there are many very smooth numbers with very few non-zero binary digits, using bounds of character sums and combinatorial arguments.

Contribution

It introduces new bounds and combinatorial techniques to show the abundance of such numbers, combining smoothness and binary digit constraints.

Findings

Many smooth numbers have very few non-zero binary digits.

The methods involve bounds of character sums and combinatorial arguments.

The results contribute to understanding the distribution of special integers.

Abstract

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