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} − DY_{1}^{2} = 1; and put ρ(D) = log X_{1}/ log X_{0}. We prove here that ρ(D) can be arbitrarily large. Indeed, we exhibit an infinite family of values of D for which ρ(D) ≫ D^{1/6}/ log D. We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which ρ(D) ≫ \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.