Consecutive square-free numbers of a special form

TL;DR

This paper proves the existence of infinitely many pairs of consecutive square-free numbers of a specific form within certain bounds, and provides an asymptotic estimate for their distribution.

Contribution

It establishes the infinitude of consecutive square-free numbers of the form [n^c], [n^c]+1 for fixed c between 1 and 7/6, with an asymptotic formula.

Findings

Infinitely many such pairs exist for 1<c<7/6.

An asymptotic formula describes their distribution.

The results extend understanding of square-free numbers in special forms.

Abstract

In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.