## Experimental Mathematics

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

### 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