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October 2000 The size of the fundamental solutions of consecutive Pell equations
Michael J. Jacobson Jr., Hugh C. Williams
Experiment. Math. 9(4): 631-640 (October 2000).

Abstract

Let D be a positive integer such that D and $D{-}1$ are not perfect squares; denote by $X_0$, $Y_0$, $X_1$, $Y_1$ the least positive integers such that $X_0^2 - (D{-}1) Y_0^2 = 1$ and $X_1^2 - D Y_1^2 = 1$; and put $\rho(D) = \log X_1 / \log X_0$. We prove here that $\rho(D)$ can be arbitrarily large. Indeed, we exhibit an infinite family of values of D for which $\rho(D) \gg D^{1/6}/ \log D$. We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which $\rho(D) \gg \sqrt{D} \log \log D / \log D$, and that this is the best possible result under the Extended Riemann Hypothesis. Finally, we present some numerical evidence in support of this heuristic.

Citation

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Michael J. Jacobson Jr.. Hugh C. Williams. "The size of the fundamental solutions of consecutive Pell equations." Experiment. Math. 9 (4) 631 - 640, October 2000.

Information

Published: October 2000
First available in Project Euclid: 20 February 2003

zbMATH: 0961.11007
MathSciNet: MR1806298

Subjects:
Primary: 11R11
Secondary: 11D09 , 11R27 , 11Y50

Keywords: continued fractions , Pell equation , read quadratic field

Rights: Copyright © 2000 A K Peters, Ltd.

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Vol.9 • No. 4 • October 2000
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