Experimental Mathematics

The size of the fundamental solutions of consecutive Pell equations

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


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

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

First available in Project Euclid: 20 February 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Pell equation continued fractions read quadratic field


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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