On the negative Pell equation

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.

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      author={Rédei, L.},
      TITLE={Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlk\" orper},
      JOURNAL={J. Reine Angew. Math.},
      VOLUME={171},
      PAGES={55--60},
      YEAR={1934},
      ZBLNUMBER={0009.05101},
      JFMNUMBER={60.0125.02},
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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