Mathematica Eterna
Open Access

Abstract

Solutions of the Pell Equations x2 − (a2b2 + 2b)y2 = N when N ∈ {±1, ±4}

Merve G¨uney

Let a and b be natural number and d = a 2 b 2 + 2b. In this paper, by using continued fraction expansion of √ d, we find fundamental solution of the equations x 2 − dy2 = ±1 and we get all positive integer solutions of the equations x 2 − dy2 = ±1 in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations x 2 − dy2 = ±4 in terms of generalized Fibonacci and Lucas sequences.