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arXiv · 2406.06191

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

Tobias Hilgart·Volker Ziegler
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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.

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Tobias Hilgart, Volker Ziegler. 2024-06-10. On the unique solvability of the simultaneous Pell equations $x^2-ay^2 = 1$ and $z^2-bx^2 = 1$. https://arxiv.org/abs/2406.06191

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