Notes on Hong's conjectures of real number power LCM matrices
Notes on Hong's conjectures of real number power LCM matrices
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DOI：
10.1016/j.jalgebra.2007.05.005
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发表时间：
2007-09
影响因子：
0.9
通讯作者：
Mao Li
中科院分区：
文献类型：
--
作者：
Mao Li
设e是真实的数，S={x1，.，xn}是n个不同正整数的集合.如果对于所有1 i，j n，（xi，xj）∈S（分别为[xi，xj] ∈S），则集合S被称为gcd-闭（分别为lcm-闭）。以xi和xj的最小公倍数的th次幂[公式：见正文]作为其第i，j项的矩阵称为th次幂最小公倍数（LCM）矩阵，用[公式：见正文]（或缩写为（[S]e））表示。本文证明了：对任意真实的数e ∈ 1，n ∈ 7，定义在任意gcd-闭（lcm-闭）集S={x1，.，xn}上的幂LCM矩阵[公式：见正文]是非奇异的.这部分地证实了Hong在[S.洪，与算术函数类相关的矩阵的非奇异性，J.代数281（2004）1-14]。对互反真实的幂GCD矩阵也得到了类似的结果。
Let e be a real number and S={x1,…,xn} be a set of n distinct positive integers. The set S is said to be gcd-closed (respectively lcm-closed) if (xi,xj)∈S (respectively [xi,xj]∈S) for all 1⩽i,j⩽n. The matrix having eth power [Formula: see text] of the least common multiple of xiand xjas its i,j-entry is called the eth power least common multiple (LCM) matrix, denoted by [Formula: see text] (or abbreviated by ([S]e)). In this paper, we show that for any real number e⩾1 and n⩽7, the power LCM matrix [Formula: see text] defined on any gcd-closed (respectively lcm-closed) set S={x1,…,xn} is nonsingular. This confirms partially two conjectures raised by Hong in [S. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 281 (2004) 1–14]. Similar results are established for reciprocal real number power GCD matrices.