Smooth sums with small spacings
Wouter van Doorn, Anneroos R.F. Everts

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 can be written as a sum of distinct -smooth integers with .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · semigroups and automata theory
