关于第二类Stirling数的p-adic赋值的一些新结果  

Some new results on the p-adic valuations of Stirling numbers of the second kind

作　　者：赵伟 邱敏 ZHAO Wei;QIU Min(Science and Technology on Communication Security Laboratory,Chengdu 610041,China;School of Science,Xihua University,Chengdu 610039,China)

机构地区：[1]保密通信重点实验室,成都610041 [2]西华大学理学院,成都610039

出　　处：《四川大学学报（自然科学版）》2020年第5期865-870,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(11771304)。

摘　　要：设n和k为任意正整数.第二类Stirling数,记作S(n,k),表示将n个元素划分为恰好k个非空集合的个数.设p为奇素数,令vp(n)表示n的p-adic赋值,即vp(n)是能整除n的最大的p的方幂.一般来说,计算S(n,k)的p-adic赋值是很困难的.有许多作者研究了第二类Stirling数S(n,k)的算术性质,包括Davis,Lengyel以及Hong等.在本文中,我们研究第二类Stirling数的p-adic赋值的一些性质.事实上,我们通过对S(n,k)进行p-adic分析证明了vp(S(p,2))≥1,其中等号成立当且仅当p为一个Wieferich素数.当n≥2时,我们还证明了vp(S(p^n,2p))≥n,以及vp(S(p^n,4p))≥n-2(p≥5),这改进了Adelberg不久前的结果.Let nand kbe positive integers.The Stirling number of the second kind,denoted by S(n,k),is defined as the number of ways to partition a set of nelements into exactly knonempty subsets.Let p be an odd prime,vp(n)stand for the p-adic valuation of n,i.e.,vp(n)is the biggest nonnegative integer r with p^r dividing n.It is difficult to evaluate vp(S(n,k))in general.There are many authors including Davis,Lengyel,Hong et al.who investigated vp(S(n,k)).In this paper,we consider the p-adic valuations of some special Stirling numbers of the second kind.In fact,by providing ap-adic analysis of S(n,k),we show that vp(S(p,2))≥1 with the equality holding if and only if pis a Wieferich prime.For n≥2,we also prove that vp(S(p^n,2p))≥n,and vp(S(p^n,4p))≥n-2 with p≥5,which improve the result given by Adelberg recently.

关 键 词：第二类STIRLING数 p-adic赋值 同余式 Wieferich素数 

分 类 号：O156.1[理学—数学]