2022/03/10 by Simone Dutto, Dutto, Simone, Nadir Murru +1
Computer Science · Mathematics · #11D25 #11D45 #11D79 #12E20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algorithms and Data Compression #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2203.05290
openalex publication_date 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Pell equation can be extended to the cubic case considering the elements of norm one in Z[√[3]r], which satisfy x3 + r y3 + r2 z3 - 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.