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Ambiguous solutions of a Pell equation

Amir Akbary and Forrest J. Francis

Vol. 16 (2023), No. 1, 13–25
Abstract

It is known that if the negative Pell equation X2 DY 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 (xn,yn) are given by the formula

xn + ynD = ±(x + yD)2n+1

for n . Furthermore, a theorem of Walker from 1967 states that if the equation aX2 bY 2 = ±1 is solvable, and if (x,y) is its solution with the smallest positive x and y, then all of its solutions (xn,yn) are given by

xna + ynb = ±(xa + yb)2n+1

for n . 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 X2 DY 2 = ±N. We also provide a criterion for determination of the nontrivial ambiguous classes of the solutions of Pell equations.

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