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Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 130–136 | Cite as

On solutions of the simultaneous Pell equations \( x^{2}-\left( a^{2}-1\right) y^{2}=1\) and \(y^{2}-pz^{2}=1\)

  • Nurettin IrmakEmail author
Article

Abstract

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations
$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$
assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).

Keywords

Diophantine equation Pell equation Recurrence 

Mathematics Subject Classification

11D61 11B39 

Notes

Acknowledgments

The author expresses his gratitude to the anonymous reviewer for the instructive suggestions.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Mathematics Deparment, Art and Science FacultyNiğde UniversityNiğdeTurkey

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