Powered numbers in short intervals II

TL;DR

This paper improves results on the distribution of powered numbers within short intervals, utilizing advanced sieve techniques, polynomial identities, and recent progress on arithmetic progressions, both unconditionally and assuming the abc-conjecture.

Contribution

It introduces new bounds and methods for analyzing powered numbers in short intervals, leveraging recent breakthroughs and classical tools.

Findings

Enhanced bounds for powered numbers in short intervals

Application of sieve methods and polynomial identities

Conditional results based on the abc-conjecture

Abstract

In this article, we derive better results concerning powered numbers in short intervals, both unconditionally and conditionally on the $abc$-conjecture. We make use of sieve method, a polynomial identity, and a recent breakthrough result on density of sets with no $k$-term arithmetic progression. In the process, we study integers over short intervals that have with a big smooth divisor.