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Mathematics of Computation

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Solving constrained Pell equations

Author: Kiran S. Kedlaya
Journal: Math. Comp. 67 (1998), 833-842
MSC (1991): Primary 11Y50; Secondary 11D09, 11D25
MathSciNet review: 1443123
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Abstract: Consider the system of Diophantine equations $x^2 - ay^2 = b$, $P(x,y) = z^{2}$, where $P$ is a given integer polynomial. Historically, such systems have been analyzed by using Baker's method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases $P(x, y) = cy^2 + d$ and $P(x, y) = cx + d$, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

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Additional Information

Kiran S. Kedlaya
Affiliation: Department of Mathematics Princeton University Princeton, New Jersey 08544

Keywords: Pell equations, integer points on elliptic curves, computer solution of Diophantine equations
Received by editor(s): January 11, 1995
Received by editor(s) in revised form: November 4, 1996
Additional Notes: This work was done during a summer internship at the Supercomputing Research Center (now Center for Computing Studies), Bowie, MD, in the summer of 1992.
Article copyright: © Copyright 1998 American Mathematical Society