关于1,1/3s1,...,1/(2n-1)sn-1的第二类初等对称函数的整性  

On the integrality of the second elementary symmetric function of 1,1/3s1,...,1/(2n-1)sn-1

作　　者：冯雨露 赵伟 FENG Yu-Lu;ZHAO Wei(College of Mathematics,Sichuan University,Chengdu 610064,China;Science and Technology on Communication Security Laboratory,Chengdu 610041,China)

机构地区：[1]四川大学数学学院,成都610064 [2]保密通信重点实验室,成都610041

出　　处：《四川大学学报（自然科学版）》2020年第3期431-434,共4页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(11771304)。

摘　　要：设d,m与n均为正整数.1915年, Theisinger证明当n≥2时,n次调和和1+1/2+...+1/n不是一个整数.1946年,Erd?s和Niven证明仅有有限多个n,使得关于1/m, 1/(m+d),…, 1/(m+nd)的一个或多个初等对称函数是整数.2015年,Wang和Hong证明当n≥2时,关于1, 1/3,..., 1/(2n-1)的所有初等对称函数均非整数.本文证明:如果n≥2,那么对任意n维正整数向量Sn=(s0,s1,...,sn-1),1, 1/3s1,..., 1/(2n-1)sn-1的第二类初等对称函数H2(Sn)=■不是一个整数.Let d,m and n be positive integers. In 1915, Theisinger proved that if n≥2, then the n-th harmonic sum 1+1/2+...+1/n is not an integer. In 1946, Erd?s and Niven extended Theisinger′s theorem by showing that there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1/m, 1/(m+d),..., 1/(m+nd) are integers. In 2015, Wang and Hong proved that none of the elementary symmetric functions of 1, 1/3,...,1/(2n-1) is an integer if n≥2. In this paper, we show that if n≥2, then for arbitrary sequence Sn=(s0,s1,...,sn-1) of positive integers(note that all the si are not necessarily distinct and not necessarily monotonic), the following reciprocal power sum H2(Sn)=■is never an integer.

关 键 词：初等对称函数 整性 p-adic赋值 

分 类 号：O156.1[理学—数学]