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 and be positive integers with . We shall call the simultaneous Diophantine equations

*simultaneous Pell equations in and .* Each such pair has the trivial solution but some pairs have nontrivial solutions too. For example, if and , then 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 .

**1**A. Baker and H. Davenport,*The equations 3𝑥²-2=𝑦² and 8𝑥²-7=𝑧²*, Quart. J. Math. Oxford Ser. (2)**20**(1969), 129–137. MR**0248079****2**Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery,*An introduction to the theory of numbers*, 5th ed., John Wiley & Sons, Inc., New York, 1991. MR**1083765****3**R. G. E. Pinch,*Simultaneous Pellian equations*, Math. Proc. Cambridge Philos. Soc.**103**(1988), no. 1, 35–46. MR**913448**, 10.1017/S0305004100064598**4**C. L. Siegel,*Über einige Anwendungen diophantischer Approximationen*, Abh. Preuss. Akad. Wiss. 1929.**5**Michel Waldschmidt,*A lower bound for linear forms in logarithms*, Acta Arith.**37**(1980), 257–283. MR**598881**

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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

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