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Simultaneous Pell Equations

Author(s): W. S. Anglin.
Journal: Math. Comp. 65 (1996), 355-359.
MSC (1991): Primary 11D09
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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 and . 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 $R<S\le 200$ .

References:

1: A. Baker and H. Davenport, The equations and , Quart. J. Math. Oxford Ser. (2) 20 (1969), 129--137. MR 40:1333
2: I. Niven, H. Zuckerman, and H. Montgomery, An introduction to the theory of numbers, 5th ed., Wiley, New York, 1991. MR 91i:11001
3: R. G. E. Pinch, Simultaneous Pellian equations, Math. Proc. Cambridge Philos. Soc. 103 (1988), 35--46. MR 89a:11029
4: C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929.
5: M. Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257--283. MR 82h:10049


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