Year: 2025
Type: article
Abstract: We prove that, for all primes $ p > 5 $, positive integers $ r $ and $ m $ with $ p\nmid m $ and $ p\nmid \binom{2mp^{r-1}}{mp^{r-1}} $, there holds$ \begin{align*} &\frac{1}{m^4p^{4r}{\binom{2mp^{r-1}}{mp^{r-1}}}^3}\bigg(\sum\limits_{k = 0}^{mp^r-1}(21k+8){\binom{2k}{k}}^3-p\sum\limits_{k = 0}^{mp^{r-1}-1}(21k+8){\binom{2k}{k}}^3\bigg)\\ &\quad \equiv-6\frac{H_{p-1}}{p^{2}}\ ({\rm{mod}}\ {p^{2}}), \end{align*} $where $ H_n = \sum_{j = 1}^{n}\frac{1}{j} $ ($ n = 0, 1, 2, \cdots $) are the usual harmonic numbers. This partially confirms a conjecture by Z.-W. Sun.
Source: AIMS Mathematics
DOI: 10.3934/math.2025521
Authors: Wenbin Zhang, Yong Zhang, Jiachen Wu
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,200
Award: G6683270681
OpenAlex ID: W4410575819