Smooth squarefree and square-full integers in arithmetic progressions

TL;DR

This paper establishes new lower bounds for the distribution of smooth squarefree integers in arithmetic progressions and estimates the minimal squarefull numbers in most residue classes modulo a prime, advancing understanding in number theory.

Contribution

It provides improved bounds on smooth squarefree integers in residue classes and estimates the smallest squarefull numbers in most classes, extending previous results.

Findings

New lower bounds for smooth squarefree integers in residue classes.

Estimates for the smallest squarefull numbers in most residue classes.

Results improve upon earlier bounds by Balog and Pomerance in certain ranges.

Abstract

We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.