On Mersenne Power GCD Matrices

Abstract

This paper explores the ???? × ???? Mersenne power GCD matrices defined on sets of positive integers, focusing on factor-closed and gcd-closed sets. By employing the form ????(???????? ,????????) = 2 (???????? ,???????? ) − 1, we investigate the ???? ????ℎ power Mersenne GCD matrix (???????? ) and provide comprehensive insights into its factorizations, determinants, reciprocals, and inverses. Building upon previous research, particularly Chun's work on power GCD matrices, we extend the analysis to Mersenne numbers, offering a thorough understanding of their properties. The study contributes to the broader understanding of arithmetic functions and their applications in matrix theory.