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### Amir Akbary and Forrest J. Francis

Vol. 16 (2023), No. 1, 13–25
##### Abstract

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

 ${x}_{n}+{y}_{n}\sqrt{D}=±{\left(x+y\sqrt{D}\right)}^{2n+1}$

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

 ${x}_{n}\sqrt{a}+{y}_{n}\sqrt{b}=±{\left(x\sqrt{a}+y\sqrt{b}\right)}^{2n+1}$

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 ${X}^{2}-D{Y}^{2}=±N$. 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
Primary: 11D09