Arithmetic Progressions in Squarefull Numbers

TL;DR

This paper investigates the existence of arithmetic progressions within k-full numbers, providing new results and constraints, especially under the assumption of the abc-conjecture, thus advancing understanding of their distribution.

Contribution

It answers Erdős's questions on arithmetic progressions in k-full numbers and derives new constraints assuming the abc-conjecture.

Findings

Existence of arithmetic progressions in k-full numbers addressed.

Derived arithmetic constraints under abc-conjecture.

Extended understanding of distribution of k-full numbers.

Abstract

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterl\'e.