arXiv subjects

Jianrong Zhao

Publications and source records attributed to Jianrong Zhao.

5 recordsLinked to original sources

arXiv2025

Studying the divisibility of power LCM matrics by power GCD matrices on gcd-closed sets

Let $S=\{x_1,\ldots, x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S $ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets $S$ with $|S|\ge 4$ such that the GCD matrix $(S)$ divides the LCM matrix $[S]$ in the ring $M_{n}(\mathbb{Z})$. For $x\in S,$ let $G_S(x):=\{z\in S: z<x, z|x \text{ and } (z|y|x, y\in S)\Rightarrow y\in\{z,x\}\}$. In 2009, Feng, Hong and Zhao answered this problem in the context where $\max_{x \in S}\{|G_S(x)|\} \leq 2$. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set $S$ with $\max_{x \in S}\{|G_S(x)|\}=3$ such that $(S)|\left[S\right].$ Meanwhile, they raised a conjecture on the necessary and sufficient condition such that $(S)|\left[S\right]$ holds for the remaining case $\max_{x \in S}\{|G_S(x)|\}\ge 4$. In this papar, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.

math.CO↗

arXiv2014

The 2-adic valuations of differences of Stirling numbers of the second kind

Let $m, n, k$ and $c$ be positive integers. Let $ν_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $ν_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $ν_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.

math.NT↗

arXiv2014

Divisibility by 2 of Stirling numbers of the second kind and their differences

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $ν_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind. It is shown that $ν_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1,$ where $0 4$ is a power of 2, and $δ(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1.

math.NT↗

arXiv2012

The universal Kummer congruences

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.

math.NT↗

arXiv2011

The 2-adic valuations of Stirling numbers of the second kind

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.

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