Solutions of the Pell equations x^2-(a^2+2a)y^2=N via generalized Fibonacci and Lucas numbers

TL;DR

This paper derives solutions to specific Pell equations with d=a^2+2a using continued fractions and generalized Fibonacci and Lucas sequences, providing explicit formulas for solutions when N is -1, +1, -4, or +4.

Contribution

It introduces a novel approach to solving Pell equations with d=a^2+2a by expressing solutions through generalized Fibonacci and Lucas numbers.

Findings

Explicit formulas for solutions using continued fractions.

Representation of solutions via generalized Fibonacci and Lucas sequences.

Solutions for N in {-1, +1, -4, +4} cases.

Abstract

In this study, we find continued fraction expansion of sqrt(d) when d=a^2+2a where a is positive integer. We consider the integer solutions of the Pell equation x^2-(a^2+2a)y^2=N when N={-1,+1,-4,+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}) via the generalized Fibonacci and Lucas sequences.