Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Abstract

Let a, n ⩾ 1 be integers and S = { x 1, … , x n } be a set of n distinct positive integers. The matrix having the ath power ( x i , x j ) a of the greatest common divisor of x i and x j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by ( S a ). Similarly we can define the ath power LCM matrix [ S a ]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S 1 ∪ ⋯ ∪ S k , where k ⩾ 1 is an integer and all S i (1 ⩽ i ⩽ k) are divisor chains such that (max( S i ), max( S j )) = gcd( S) for 1 ⩽ i ≠ j ⩽ k. In this paper, we first obtain formulae of determinants of power GCD matrices ( S a ) and power LCM matrices [ S a ] on the set S consisting of finitely many quasi-coprime divisor chains with gcd( S) ∈ S. Using these results, we then show that det( S a )∣det( S b ), det[ S a ]∣det[ S b ] and det( S a )∣det[ S b ] if a∣ b and S consists of finitely many quasi-coprime divisor chains with gcd( S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.