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Solution of certain Pell equations

Zahid Raza

http://orcid.org/0000-0002-4910-2038

Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates

E-mail Address: zraza@sharjah.ac.ae

Corresponding author.

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and

Hafsa Masood Malik

Department of Mathematics, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore, Pakistan

E-mail Address: hafsa.masood.malik@gmail.com

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https://doi.org/10.1142/S1793557118500560Cited by:1 (Source: Crossref)

Abstract

Let $a, b, c$ be any positive integers such that $c | a b$ and $d_{i}^{\pm}$ is a square free positive integer of the form $d_{i}^{\pm} = a^{2 k} b^{2 l} \pm i c^{m}$ where $k, l \geq m$ and $i = 1, 2 .$ The main focus of this paper is 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 $F_{Δ_{d_{i}^{\pm}}} (x, y) = 1$ in terms of $d_{i}^{\pm} .$ We extend all the results of the papers [3, 10, 27, 37].

Communicated by G. O. H. Katona

Keywords:

AMSC: 11D09, 11D79, 11D45, 11A55, 11B39, 11B50

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