On the number of $k$-full integers between three successive $k$-th powers

Abstract

Let $k\ge 2$ be an integer. The aim of this paper is to investigate the distribution of $k$-full integers between three successive $k$-th powers. More precisely, for any integers $\ell ,m\ge 0$, we establish the explicit asymptotic density for the set of integers $n$ such that the intervals $(n^k,(n+1{)}^k)$ and $((n+1{)}^k,(n+2{)}^k)$ contain exactly $\ell$ and $m$ $k$-full integers, respectively. As an application, we prove that there are infinitely many triples of successive $k$-th powers in the sequence of $k$-full integers, thereby providing a more general answer to Shiu's question.

14 pages