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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Diophantine equation $b^2X^4-dY^2=1$
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by Michael A. Bennett and Gary Walsh PDF
Proc. Amer. Math. Soc. 127 (1999), 3481-3491 Request permission

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
  • MR Author ID: 339361
  • 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
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3481-3491
  • MSC (1991): Primary 11D25, 11J86
  • DOI: https://doi.org/10.1090/S0002-9939-99-05041-8
  • MathSciNet review: 1625772