Solving constrained Pell equations
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- by Kiran S. Kedlaya PDF
- Math. Comp. 67 (1998), 833-842 Request permission
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.References
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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
- 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.
- © Copyright 1998 American Mathematical Society
- 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