Greatest common divisor matrices

Abstract

Let S={ x 1, x 2,…, x n } be a set of distinct positive integers. Then n × n matrix [ S]=( S ij ), where S ij =( x i , x j ), the greatest common divisor of x i and x j , is call the greatest common divisor (GCD) matrix on S. We initiate the study of GCD matrices in the direction of their structure, determinant, and arithmetic in Z n . Several open problems are posed.