Experimental Mathematics

The size of the fundamental solutions of consecutive Pell equations

Michael J. Jacobson, Jr. and Hugh C. Williams

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.

Article information

Source
Experiment. Math., Volume 9, Issue 4 (2000), 631-640.

Dates
First available in Project Euclid: 20 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1045759528

Mathematical Reviews number (MathSciNet)
MR1806298

Zentralblatt MATH identifier
0961.11007

Subjects
Primary: 11R11: Quadratic extensions
Secondary: 11D09: Quadratic and bilinear equations 11R27: Units and factorization 11Y50: Computer solution of Diophantine equations

Keywords
Pell equation continued fractions read quadratic field

Citation

Jacobson, Michael J.; Williams, Hugh C. The size of the fundamental solutions of consecutive Pell equations. Experiment. Math. 9 (2000), no. 4, 631--640. https://projecteuclid.org/euclid.em/1045759528


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