Positive Integer Solutions of the Pell Equation $x^{2}-dy^{2}=N,$ $% d\in \left\{k^{2}\pm 4,\text{}k^{2}\pm 1\right\} $ and $N\in \left\{\pm 1,\pm 4\right\}

TL;DR

This paper investigates positive integer solutions to Pell equations with specific values of d and N, using continued fractions and expressing solutions through generalized Fibonacci and Lucas sequences, offering an elementary and novel approach.

Contribution

It introduces an elementary method using continued fractions to find solutions for Pell equations with d in {k^2±1, k^2±4} and N in {±1, ±4}, expressing solutions via generalized sequences.

Findings

Derived fundamental solutions using continued fractions.

Expressed all solutions in terms of generalized Fibonacci and Lucas sequences.

Provided an elementary alternative method to existing solutions.

Abstract

Let $\ k$ be a natural number and $d=k^{2}\pm 4$ or $k^{2}\pm 1$. In this paper, by using continued fraction expansion of $\sqrt{d},$ we find fundamental solution of the equations $x^{2}-dy^{2}=\pm 1$ and we get all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 1$ in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 4$ in terms of generalized Fibonacci and Lucas sequences. Although some of the results are well known, we think our method is elementary and different from the others.