# On the digital representation of smooth numbers

## TL;DR

This paper investigates the properties of smooth numbers in different bases, showing that large integers with limited prime factors and digits in base $b$ are rare unless divisible by $b$, highlighting their structural constraints.

## Contribution

It provides a quantitative analysis of the prime factorization and digit representation constraints of large integers in a given base, revealing new limitations on their structure.

## Key findings

- Large integers not divisible by $b$ cannot have both few prime factors and few nonzero digits in base $b$
- Quantitative bounds on the number of prime factors and nonzero digits for such integers
- Structural restrictions on the digital representation of smooth numbers

## Abstract

Let $b \ge 2$ be an integer. Among other results, we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of $b$ cannot have simultaneously only few distinct prime factors and only few nonzero digits in its representation in base $b$.