Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains
Let \(a,n\geq 1\) be integers, \(S=\{x_1,\dots,x_n\}\) a set of \(n\) distinct positive integers. The matrix having the \(a\)th power \((x_i,x_j)^a\) of the greatest common divisor of \(x_i\) and \(x_j\) as its \(i,j\)-entry is called \(a\)th power greatest common divisor (GCD) matrix defined on \(S\), denoted by \((S^a)\). Similarly one defines the \(a\)th power LCM matrix \( [S^a]\). The authors show that \(\det(S^a)\mid \det(S^b)\), \(\det[S^a]\mid \det[S^b] \) and \(\det(S^a)\mid \det[S^b]\) if \(a\mid b\) and \(S\) consists of finitely many quasi-coprime divisor chains with \(\gcd(S)\in S\).
scientific article; zbMATH DE number 5836683
scientific article