A note on powerful numbers in short intervals

TL;DR

This paper investigates upper bounds for the count of powerful numbers within short intervals, providing unconditional and conditional bounds based on the $abc$-conjecture, and relating these to classical number theory conjectures.

Contribution

It establishes new uniform upper bounds for powerful numbers in short intervals, both unconditionally and assuming the $abc$-conjecture, advancing understanding of their distribution.

Findings

Unconditional bounds: $O(rac{y}{\log y})$ and $O(y^{11/12})$ for all and smooth powerful numbers.

Conditional bounds: $O(rac{y}{\log^{1+\epsilon} y})$ and $O(y^{(2 + \epsilon)/k})$ for $k$-full numbers.

Connections to Roth's theorem and the conjecture on three consecutive squarefull numbers.

Abstract

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{\log^{1+\epsilon} y})$ for squarefull numbers and the bound $O(y^{(2 + \epsilon)/k})$ for $k$-full numbers when $k \ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numbers.