Solving constrained Pell equations

Author:
Kiran S. Kedlaya

Journal:
Math. Comp. **67** (1998), 833-842

MSC (1991):
Primary 11Y50; Secondary 11D09, 11D25

DOI:
https://doi.org/10.1090/S0025-5718-98-00918-1

MathSciNet review:
1443123

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Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
349028

ORCID:
0000-0001-8700-8758

Email:
kkedlaya@math.princeton.edu

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