The number of real quadratic fields having units of negative norm

Abstract

We study the density of the set of real quadratic fields for which the norm of the fundamental unit equals $-$1 inside the set of real quadratic fields containing elements of norm $-$1. A conjectural density is derived from a single heuristic assumption, and experimental data supporting this assumption are given. We finally discuss how close one can get to proving such conjectural densities.