Gauss’s one-class-per-genus problem for negative discriminants

Determine whether Gauss’s list of 65 negative discriminants Δ ≡ 0 (mod 4) with one class per genus is complete; equivalently, classify all negative discriminants Δ ≡ 0 (mod 4) for which the primitive positive definite integral binary quadratic forms of discriminant Δ have one class per genus.

Background

The paper reviews the classical one-class-per-genus problem introduced by Gauss for positive definite binary quadratic forms of negative discriminant. Gauss produced an empirical list of 65 such discriminants and asked if the list is complete.

This longstanding problem remains open. The authors discuss it to contextualize their real quadratic analogue involving unit-generated orders and genus theory, highlighting the difficulty of achieving complete classifications in related settings.