On Mersenne GCED Matrices

Abstract

A Mersenne number is defined as a number of the form M_n=2^n-1, where n is a positive integer. The first five Mersenne numbers are 1, 3, 7, 15, and 31. A divisor d of a positive integer m=p^k, where p is a prime, is termed an exponential divisor if it satisfies d=p^t with t dividing k, and it is denoted as d|_e m. Two integers a and b share a common exponential divisor if they have the same prime factors. The greatest common exponential divisor (GCED) of two integers a and b is denoted by gced(a, b). A set S is called exponential factor-closed if the exponential divisor of every element of S also belongs to S. Similarly, S is GCED-closed if gced(a, b) belongs to S for every pair a,b in S. If S is an exponential factor-closed set of distinct positive integers arranged in increasing order, the GCED matrix associated with S is the matrix M, where each entry M_ij is given by gced(a_i,a_j). The Mersenne GCED matrix M associated with S is a square matrix where each entry M_ij is of the form gced(2^(a_i )-〖1,2〗^(a_j )-1). This paper introduces the concept of Mersenne GCED square matrices defined on a non-exponential factor-closed set. We establish a comprehensive characterization of their fundamental properties, including their structure, determinant, reciprocal, and inverse.