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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 $R<S\le 200$.

References [Enhancements On Off] (What's this?)

  • 1 A. Baker and H. Davenport, The equations $3x^2 - 2 = y^2$ and $8x^2 - 7 = z^2$, 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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Additional Information

W. S. Anglin
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montreal, Quebec, Canada H3A 2K6

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

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