The 3-adic valuations of Stirling numbers of the first kind
Abstract
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)) $. This proves the case $ p = 3 $ of a conjecture of Hong and Qiu proposed in 2020. As applications, we derive formulas near the diagonal, comparison results for adjacent orders, 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.
Fields of Study & Subjects
Advanced Mathematical Identities
Advanced Combinatorial Mathematics
Analytic Number Theory Research
Publication details
- Journal / source
- AIMS Mathematics
- Volume
- 11
- Issue
- 8
- Pages
- 25553–25600
- Publication date
- 1 January 2026
- Language
- EN