On the number of $k$-full integers between three successive $k$-th powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915803120533504 |
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| author | Narumi, Shusei Tachiya, Yohei |
| author_facet | Narumi, Shusei Tachiya, Yohei |
| contents | Let $k\geq2$ 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 $\ell,m\ge0$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of $k$-full integers between three successive $k$-th powers Narumi, Shusei Tachiya, Yohei Number Theory 11N25, 11N64, 11J71 Let $k\geq2$ 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 $\ell,m\ge0$, 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. |
| title | On the number of $k$-full integers between three successive $k$-th powers |
| topic | Number Theory 11N25, 11N64, 11J71 |
| url | https://arxiv.org/abs/2512.07438 |