Density computations for real quadratic units

In order to study the density of the set of positive integers $d$ for which the negative Pell equation $x^{2}-dy^{2}=-1$ is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in $\log d$ as the factorization of $d$ is a natural part of the input for the values of $d$ we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion $P=1-\prod _{j\ge 1 \text {\rm odd}} (1-2^{-j}) \approx .58$ of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm $-1$.