Cubic Pell’s Equation

This is an initial study of Pell’s equations of higher degree, which is an open problem in Number Theory. The rst step is to investigate the Pell’s equation of the form x³-dy³ = 1. Later, we consider the form N(θ) = x³ + cy³ + c²z³ – 3cxyz = 1, where θ³ = c for some non-perfect cube integer c. For this form, it is found that for some certain c values, solutions can be generated by an algorithm similar to that for the quadratic Pell case. However, this algorithm does not work for all c values, for example c = 15 and c = 16. Investigating these equations involves literature studies and computational researches in Wolfram Mathematica as well as MatLab R2011a. The cubic Pell’s equation has possible applications within Approximation Theory and Cryptography.

Tien Truong

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