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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Simultaneous Pell Equations


Author: W. S. Anglin
Journal: Math. Comp. 65 (1996), 355-359
MSC (1991): Primary 11D09
MathSciNet review: 1325861
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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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Additional Information

W. S. Anglin
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montreal, Quebec, Canada H3A 2K6
Email: anglin@math.mcgill.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00687-4
PII: S 0025-5718(96)00687-4
Keywords: Diophantine, Pell
Received by editor(s): June 8, 1994
Received by editor(s) in revised form: October 11, 1994
Article copyright: © Copyright 1996 American Mathematical Society