Class number one for unit-generated non-maximal quadratic orders

Determine all discriminants Δ of the form n^2 ± 4 such that the associated real quadratic unit-generated order O_Δ has class number h_Δ = 1 in the non-maximal case (i.e., when Δ is not a fundamental discriminant or equivalently the conductor f_Δ > 1).

Background

The paper completely classifies unit-generated maximal orders of real quadratic fields with class number one. For non-maximal orders, the authors provide computational data up to Δ < 10{10} and note that the list is complete for fundamental discriminants, but the completeness for non-fundamental discriminants remains unresolved.

This problem aligns with Problem 1 stated earlier in the paper (“Determine all unit-generated quadratic orders O_Δ having h_Δ=1”) and is the remaining open part after resolving the maximal case. It is complicated by the fact that the associated maximal order to a unit-generated order need not itself be unit-generated, expanding the search space.

References

The list is complete for all fundamental discriminants (without the restriction Δ < 10{10}). It is not known to be complete for all non-fundamental discriminants.

— Unit-generated orders of real quadratic fields I. Class number bounds (2512.11311 - Kopp et al., 12 Dec 2025) in Caption of Table 1, Section 4.2