Ambiguous solutions of a Pell equation

Akbary, Amir; Francis, Forrest J.

doi:10.2140/involve.2023.16.13

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Ambiguous solutions of a Pell equation

Amir Akbary and Forrest J. Francis

Vol. 16 (2023), No. 1, 13–25

DOI: 10.2140/involve.2023.16.13

Abstract

It is known that if the negative Pell equation $X^{2} - D Y^{2} = - 1$ is solvable (in integers), and if $(x, y)$ is its solution with the smallest positive $x$ and $y$ , then all of its solutions $(x_{n}, y_{n})$ are given by the formula

x_{n} + y_{n} \sqrt{D} = \pm {(x + y \sqrt{D})}^{2 n + 1}

for $n \in ℤ$ . Furthermore, a theorem of Walker from 1967 states that if the equation $a X^{2} - b Y^{2} = \pm 1$ is solvable, and if $(x, y)$ is its solution with the smallest positive $x$ and $y$ , then all of its solutions $(x_{n}, y_{n})$ are given by

x_{n} \sqrt{a} + y_{n} \sqrt{b} = \pm {(x \sqrt{a} + y \sqrt{b})}^{2 n + 1}

for $n \in ℤ$ . We prove a unifying theorem that includes both of these results as special cases. The key observation is a structural theorem for the nontrivial ambiguous classes of the solutions of the (generalized) Pell equations $X^{2} - D Y^{2} = \pm N$ . We also provide a criterion for determination of the nontrivial ambiguous classes of the solutions of Pell equations.

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Keywords

generalized Pell equation, ambiguous classes of solutions

Mathematical Subject Classification

Primary: 11D09

Milestones

Received: 2 June 2021

Revised: 23 March 2022

Accepted: 22 April 2022

Published: 14 April 2023

Communicated by Filip Saidak

Authors

Amir Akbary
Department of Mathematics and Computer Science University of Lethbridge Lethbridge, AB Canada

Forrest J. Francis
Department of Mathematics and Computer Science University of Lethbridge Lethbridge, AB Canada	School of Science UNSW Canberra Australia

Vol. 16, No. 1, 2023