Solution of Certain Pell Equations

Let \(a,b,c \) be any positive integers such that \(c\mid ab\) and \(d_i^\pm\) is a square free positive integer of the form \(d_i^\pm=a^{2k} b^{2l}\pm i c^m\) where \(k,l \geq m\) and \(i=1,2.\) The main focus of this paper to find the fundamental solution of the equation \( x^2-d_i^\pm y^2=1\) wit...

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Bibliographic Details
Published in:arXiv.org 2014-02
Main Authors: Raza, Zahid, Hafsa Masood Malik
Format: Article
Language:English
Subjects:
Automorphisms
Integers
Mathematical analysis
Online Access:Get full text
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Summary:Let \(a,b,c \) be any positive integers such that \(c\mid ab\) and \(d_i^\pm\) is a square free positive integer of the form \(d_i^\pm=a^{2k} b^{2l}\pm i c^m\) where \(k,l \geq m\) and \(i=1,2.\) The main focus of this paper to find the fundamental solution of the equation \( x^2-d_i^\pm y^2=1\) with the help of the continued fraction of \(\sqrt{d_i^\pm}.\) We also obtain all the positive solutions of the equations \( x^2-d_i^\pm y^2=\pm 1\) and \( x^2-d_i^\pm y^2=\pm 4\) by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form \( F_{d_i^\pm}(x, y) = x^2-d_i^\pm y^2 \) including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation \( F_{\Delta_{d_i^\pm}} (x, y) = 1 \) in terms of $d_i^\pm. We generalized all the results of the papers [2], [9], [26], and [37].
ISSN:2331-8422