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Shifted powers in binary recurrence sequences

Published online by Cambridge University Press:  08 January 2015

MICHAEL A. BENNETT ,
SANDER R. DAHMEN ,
MAURICE MIGNOTTE  and
SAMIR SIKSEK
MICHAEL A. BENNETT
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada. e-mail: bennett@math.ubc.ca
SANDER R. DAHMEN
Affiliation:
Department of Mathematics, VU University Amsterdam, Amsterdam, The Netherlands. e-mail: s.r.dahmen@vu.nl
MAURICE MIGNOTTE
Affiliation:
Department of Mathematics, University of Strasbourg, Strasbourg, France. e-mail: mignotte@math.u-strasbg.fr
SAMIR SIKSEK
Affiliation:
Mathematics Institute, University of Warwick, Coventry. e-mail: S.Siksek@warwick.ac.uk
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Abstract

Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 305 - 329
Copyright
Copyright © Cambridge Philosophical Society 2015 

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