On Construction Structures of Matrix Solutions of Exponential Diophantine Equations

Abstract

We show that the matrix exponential Diophantine equation (Xn - Iqxn)(Yn - Iqxn) = Z2; admits at least 4 x n2 different construction structures of matrix solutions. We also prove that the matrix exponential Diophantine equation (Xn - Inxm)(Ym - Inxm) = Z2; admits at least 4 x n x m different construction structures of matrix solutions in Mnxm(\(\mathbb{N}\)) for every pair (n,m) of positive integers such that n \(\neq\) m. We show the connection between the construction structures of matrix solutions of an exponential Diophantine equation and Integer factorization. We show that the matrix Diophantine equation Xn +Ym = Zq , n, m, q \(\varepsilon\) \(\mathbb{N}\); admits at least 8 x n x m x q different construction structures of matrix solutions in Mnxmxq(\(\mathbb{N}\)).