On the Solutions of Some Equations Over Rings of Integers Modulo the Square of a Prime

Algebraic equations over finite fields and over finite rings have been of interest due to their beautiful structures, useful in applications, and links with other mathematical objects. In this paper, the equation X2 - Y2 = α is studied over the ring of integers modulo p2, where p is a prime number and α is an arbitrary constant. Through a matrix method, the solutions of this equation are given together with an explicit formula for the number of solutions.