Unit-generated orders of real quadratic fields I. Class number bounds

Unit-generated orders of a quadratic field are orders of the form $\mathcal{O}=\mathbb{Z}[\epsilon ]$, where $\epsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud–Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as $\mathcal{O}={\mathcal{O}}_n^{\pm }$ having quadratic discriminants $\mathrm{\Delta }(\mathcal{O})={\mathrm{\Delta }}_n^{+}=n^2-4$ (for $n\ge 3$) and $\mathrm{\Delta }(\mathcal{O})={\mathrm{\Delta }}_n^{-}=n^2+4$ (for $n\ge 1$). We show the (wide or narrow) class numbers of unit-generated orders satisfy $\log \mid \mathrm{C}\mathrm{l}(\mathcal{O})\mid \sim \log \frac{1}{2}\mid \mathrm{\Delta }(\mathcal{O})\mid$ as $\mid \mathrm{\Delta }(\mathcal{O})\mid \to \mathrm{\infty }$, using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is $2$-torsion. We classify all unit-generated real quadratic orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are $2$-torsion for $\mathrm{\Delta }\le {10}^{10}$, for both wide and narrow class groups. These lists are conjecturally complete for all $\mathrm{\Delta }$.