GCD和GCUD矩阵行列式的一个不等式（英）

An Inequality for the Determinant of the GCD Matrix and the GCUD Matrix

英文关键词:Srith′s determinant  GCD matrix  unitary analogue  inequality.

中文摘要:

      

英文摘要:

      Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).