Classification of unit-generated real quadratic orders with 2-torsion class group

Determine all discriminants Δ of the form n^2 ± 4 such that the associated real quadratic unit-generated order O_Δ has wide ideal class group of exponent 2, i.e., Cl(O_Δ) = Cl(O_Δ)[2], beyond the computational range Δ < 10^{10}.

Background

The authors establish that only finitely many unit-generated real quadratic orders have 2-torsion wide class group. They provide extensive computational tables of such discriminants for Δ < 10^{10}, covering both families Δ = n^2 - 4 and Δ = n^2 + 4, and identify those with one class per genus (narrow class group 2-torsion).

The completeness of these lists outside the given computational bound is unknown, and a full classification would parallel the classical one-class-per-genus problem. This problem subsumes the real quadratic analogue of the Gauss problem and remains a challenging open classification task.