Simultaneous Pell equations

Anglin, W.

doi:10.1090/S0025-5718-96-00687-4

Simultaneous Pell equations
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by W. S. Anglin PDF

Math. Comp. 65 (1996), 355-359 Request permission

Abstract:

Let $R$ and $S$ be positive integers with $R<S$. We shall call the simultaneous Diophantine equations \begin{align*} x^2-Ry^2&=1,\ z^2-Sy^2&=1 \end{align*} simultaneous Pell equations in $R$ and $S$. Each such pair has the trivial solution $(1,0,1)$ but some pairs have nontrivial solutions too. For example, if $R=11$ and $S=56$, then $(199, 60, 449)$ is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when $R<S\le 200$.

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