Jianrong Zhao

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20082025

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math.NT2014

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

Wei Zhao, Jianrong Zhao, Shaofang Hong

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 esta…

math.NT2012

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

Jianrong Zhao, Shaofang Hong, Wei Zhao

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 $k$ , respectivel…

math.NT2009

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

Shaofang Hong, Jianrong Zhao, Wei Zhao

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S (n, k)$ of the second kind. We show that $v_{2} (S (4 i, 5)) = v_{2} (S (4 i + 3, 5))$ if and only if $i\not\equiv 7…

math.NT2008

The universal Kummer congruences

Shaofang Hong, Jianrong Zhao, Wei Zhao

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…