On the cubic Pell equation over finite fields

TL;DR

This paper investigates the cubic Pell equation over finite fields, introducing new methods to count and generate solutions, thereby extending classical results to a more complex algebraic setting.

Contribution

It presents novel techniques for counting and generating solutions to the cubic Pell equation over finite fields, a problem previously lacking comprehensive methods.

Findings

Developed a method to count solutions depending on r

Provided a systematic way to generate all solutions

Extended classical Pell equation results to cubic case

Abstract

The classical Pell equation can be extended to the cubic case considering the elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3 - 3 r x y z = 1$. 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. 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.