Powers in recurrence sequences: Pell equations

Bennett, Michael

doi:10.1090/S0002-9947-04-03586-X

Powers in recurrence sequences: Pell equations
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by Michael A. Bennett PDF

Trans. Amer. Math. Soc. 357 (2005), 1675-1691 Request permission

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 \[ x^2 - Dy^{2n} =1 \] 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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