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Abstract
The classical Pell equation can be extended to the cubic case considering the elements of norm one in 
, which satisfy
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.