Notes on Hong's conjectures of real. number power LCM matrices

作者：Li Mao^*

来源：Journal of Algebra, 2007, 315(2): 654-664.

DOI：10.1016/j.jalgebra.2007.05.005

• gcd-Closed set
• 1cm-Closed set
• greatest-type divisors
• real number power LCM matrix
• reciprocal real number power GCD matrix
• nonsingularity

Let e be a real number and S = {x(1)....x(n)} be a set of n distinct positive integers. File set S is said to be gcd-closed (respectively Icm-closed) if (x(i) , x(j)) is an element of S (respectively [x(i) , x(j)]is an element of S) for all 1 <= i, j <= n. The matrix having eth power [x(i), x(j)](e) of the least common multiple of xi and xj as its i. j-entry is called the eth power least cornmon multiple (LCM) matrix, denoted by ([x(i), x(j)](e)) (or abbreviated by ([S](e))). In this paper, we show that for any real number e >= 1 and it <= 7, the power LCM matrix ([x(i), x(j)](e)) defined oil any gcd-closed (respectively Icm-closed) set S ={x(1).....,x(n)} is nonsingular. This confirms partially two conjectures raised by Hong in IS. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 28 1 (2004) 1-14]. Similar results are established for reciprocal real number power GCD matrices.

• 出版日期2007-9-15
• 单位西南大学