The Generalized Fibonacci and Lucas Solutions of The Pell Equations x^2-(a^2b^2-b)y^2=N and x^2-(a^2b^2-2b)y^2=N

TL;DR

This paper derives solutions to specific Pell equations using continued fractions and expresses these solutions through generalized Fibonacci and Lucas sequences, providing explicit formulas for the n-th solutions.

Contribution

It introduces a method to explicitly formulate solutions of Pell equations with specific parameters using continued fractions and generalized Fibonacci and Lucas sequences.

Findings

Explicit formulas for solutions of Pell equations with given parameters.

Representation of solutions in terms of generalized Fibonacci and Lucas sequences.

Derivation of continued fraction expansions for specific square roots.

Abstract

In this study, we find continued fraction expansion of sqrt(d) when d=a^2b^2-b and d=a^2b^2-2b where a and b are positive integers. We consider the integer solutions of the Pell equations x^2-(a^2b^2-b)y^2=N and x^2-(a^2b^2-2b)y^2=N when N is {+-1,+-4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) in terms of generalized Fibonacci and Lucas sequences.