On the cubic Pell equation over finite fields

The classical Pell equation can be extended to the cubic case considering the elements of norm one in Z[ √3 r], which satisfy x 3 + ry 3 + r 2 z 3 − 3rxyz = 1. 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.