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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Diophantine equation $b^2X^4-dY^2=1$


Authors: Michael A. Bennett and Gary Walsh
Journal: Proc. Amer. Math. Soc. 127 (1999), 3481-3491
MSC (1991): Primary 11D25, 11J86
Published electronically: May 6, 1999
MathSciNet review: 1625772
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Abstract: If $b$ and $d$ are given positive integers with $b > 1$, then we show that the equation of the title possesses at most one solution in positive integers $X,Y$. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.


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Additional Information

Michael A. Bennett
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: mabennet@ias.edu, mabennet@math.uiuc.edu

Gary Walsh
Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5
Email: gwalsh@mathstat.uottawa.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05041-8
PII: S 0002-9939(99)05041-8
Keywords: Diophantine equations, Pell sequences
Received by editor(s): February 17, 1998
Published electronically: May 6, 1999
Additional Notes: The first author was supported in part by NSF Grants DMS-9700837 and DMS-9304580 and through the David and Lucile Packard Foundation.
The second author was supported in part by NSERC Grant 2560150.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society