Density computations for real quadratic units

Abstract

In order to study the density of the set of positive integers d d for which the negative Pell equation x 2 − d y 2 = − 1 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 \log d as the factorization of d d is a natural part of the input for the values of d 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 o d d ( 1 − 2 − j ) ≈ .58 P = 1 - \prod _{j\ge 1\; \mathrm {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 -1 .