The Size of the Fundamental Solutions of Consecutive Pell Equations

Abstract

Let D be a positive integer such that D and D–1 are not perfect squares; denote by X0, Y0, X1, Y1 the least positive integers such that X2 0 – (D–1)Y2 0 = 1 and X2 1 – DY2 1 = 1; and put ρ(D) = log X1/log X0, We prove here that ρ(D) can be arbitrarily large. Indeed, we exhibit an infinite family of values of D for which ρ(D) » D⅙/log D. We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which ρ(D) » √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.