Smooth sums with small spacings

TL;DR

This paper proves that every positive integer can be expressed as a sum of distinct 3-smooth integers with controlled spacing, advancing understanding of additive representations involving smooth numbers.

Contribution

It establishes a universal representation of integers as sums of distinct 3-smooth numbers with bounded ratios, a new result in additive number theory.

Findings

Every positive integer can be written as a sum of distinct 3-smooth integers.

The 3-smooth integers in the sum are ordered and have ratios less than 6.

The result solves a problem posed by Erdős regarding smooth number sums.

Abstract

Solving a problem by Erd\H{o}s, we prove that every positive integer $n$ can be written as a sum $$n = b_{1} + b_{2} + \ldots + b_{r}$$ of distinct $3$-smooth integers with $1 \le b_{1} < b_{2} < \ldots < b_{r} < 6b_{1}$.