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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.13458 |
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| _version_ | 1866909050168410112 |
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| author | Qiu, Min Lin, Zongbing Chen, Long |
| author_facet | Qiu, Min Lin, Zongbing Chen, Long |
| contents | Let $v_3$ denote the usual $3$-adic valuation, and let $s(n, k)$ be the unsigned Stirling number of the first kind. In this paper, for $a\in\{1,2\}$, we determine the values of $v_3(s(a3^n, k))$ for all $1\le k\le a3^n$. More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. We also derive formulas near the diagonal, comparison results for the adjacent orders $a3^n$ and $a3^n+1$, sharp upper bounds for the families $v_3(s(3^n, k))$ and $v_3(s(2\cdot3^n, k))$, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_13458 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The 3-adic valuations of Stirling numbers of the first kind Qiu, Min Lin, Zongbing Chen, Long Number Theory Let $v_3$ denote the usual $3$-adic valuation, and let $s(n, k)$ be the unsigned Stirling number of the first kind. In this paper, for $a\in\{1,2\}$, we determine the values of $v_3(s(a3^n, k))$ for all $1\le k\le a3^n$. More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. We also derive formulas near the diagonal, comparison results for the adjacent orders $a3^n$ and $a3^n+1$, sharp upper bounds for the families $v_3(s(3^n, k))$ and $v_3(s(2\cdot3^n, k))$, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna. |
| title | The 3-adic valuations of Stirling numbers of the first kind |
| topic | Number Theory |
| url | https://arxiv.org/abs/2109.13458 |