Pages 2035-2104 from Volume 172 (2010), Issue 3 by Étienne Fouvry, Jürgen Klüners
Abstract
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.
Received: 5 July 2007
Revised: 25 July 2008
Accepted: 27 February 2008
Published online: 5 October 2010
Authors
Étienne Fouvry
Université Paris-Sud
Laboratoire de mathématique, UMR
8628
CNRS, F-91405 Orsay Cedex
France
Jürgen Klüners
Universität Paderborn
Institut für Mathematik
33095 Paderborn
Germany