On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers.)

TL;DR

This paper investigates the additive and multiplicative structure of smooth numbers, extending previous results by applying $S$-unit equations to better understand their decomposability.

Contribution

It introduces a new approach using $S$-unit equations to analyze the additive decomposability of smooth numbers, improving upon prior sieve-based methods.

Findings

Extended the results on smooth numbers' decomposability

Sharpened previous bounds and conditions

Provided new insights using $S$-unit equations

Abstract

In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).