Explicit bounds for large gaps between squarefree integers

TL;DR

This paper establishes explicit upper bounds for gaps between squarefree integers, demonstrating that intervals of a certain size always contain such integers, with potential for further improvement under larger x.

Contribution

It provides the first explicit bounds for large gaps between squarefree integers, refining previous asymptotic results with concrete constants.

Findings

Intervals of the form (x, x + 11x^{1/5} log x] always contain a squarefree integer for x ≥ 2.

The constant 11 in the bound can be reduced for sufficiently large x.

Explicit bounds improve understanding of the distribution of squarefree integers.

Abstract

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree integer. The constant 11 can be improved further, if $x$ is assumed to be larger than a (very) large constant.