On the Distribution of Cube-Free Numbers with the Form $[n^c]$

TL;DR

This paper proves that for any real number c between 1 and 11/6, there are infinitely many cube-free numbers of the form [n^c], expanding understanding of the distribution of special algebraic numbers.

Contribution

The paper establishes the infinitude of cube-free numbers of the form [n^c] for a new range of c, specifically between 1 and 11/6, which was previously unknown.

Findings

Infinitely many cube-free numbers of the form [n^c] exist for 1 < c < 11/6.

The result extends the known distribution of cube-free numbers in non-integer sequences.

The proof covers a new range of the parameter c, broadening the scope of number theory results on cube-free numbers.

Abstract

In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.