On the negative Pell equation


We give asymptotic upper and lower bounds for the number of squarefree $d$ ($0\lt d\leq X$) such that the equation $x^2-dy^2=-1$ is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm $-1$ and give strong evidence in the direction of a conjecture due to P. Stevenhagen.


Étienne Fouvry

Université Paris-Sud
Laboratoire de mathématique, UMR 8628
CNRS, F-91405 Orsay Cedex

Jürgen Klüners

Universität Paderborn
Institut für Mathematik
33095 Paderborn