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 x2 − dy2 = −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 − ∏ j≥1 odd(1 − 2−j) ≈ .58 of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm −1.