On the cubic Pell equation over finite fields

Abstract

The solution of the cubic Pell equation is harder than the classical case, indeed a method for solving it as Diophantine equation is still  missing [3]. In this paper, we study the cubic Pell equation over finite fields, extending the results that hold for the classical one. In  particular, we provide a novel method for counting the number of solutions in all possible cases depending on the value of r. Moreover,  we are also able to provide a method for generating all the solutions.