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Powers in recurrence sequences: Pell equations

Author: Michael A. Bennett
Journal: Trans. Amer. Math. Soc. 357 (2005), 1675-1691
MSC (2000): Primary 11D41; Secondary 11D45, 11B37
Published electronically: October 28, 2004
MathSciNet review: 2115381
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer $D$, we show how to (practically) solve Diophantine equations of the form

\begin{displaymath}x^2 - Dy^{2n} =1 \end{displaymath}

in integers $x, y$ and $n \geq 2$. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.

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Additional Information

Michael A. Bennett
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Keywords: Pell sequences, perfect powers, Thue equations
Received by editor(s): July 20, 2003
Received by editor(s) in revised form: December 4, 2003
Published electronically: October 28, 2004
Additional Notes: This work was supported in part by a grant from NSERC
Article copyright: © Copyright 2004 American Mathematical Society