On the unique solvability of the simultaneous Pell equations $x^2-ay^2 = 1$ and $z^2-bx^2 = 1$

TL;DR

This paper investigates the conditions under which the simultaneous Pell equations have unique solutions, providing a procedure to determine whether solutions are finite or unique for fixed parameters.

Contribution

It introduces a method to analyze the solvability of simultaneous Pell equations, identifying cases with at most one or finitely many solutions for given parameters.

Findings

Identifies conditions for unique solutions

Provides a procedure to find exceptions

Establishes finiteness of solutions in certain cases

Abstract

We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at most one solution in positive integers, or finds all exceptions for which we have proved that there are at most finitely many.