The existence of at least a vague distinction between what I shall call the constructive and the non-constructive ordinals of the second number class, that is, between the ordinals which can in some sense be effectively built up to step by step from below and those for which this cannot be done (although there may be existence proofs), is, I believe, somewhat generally recognized. My purpose he...

In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This...

Let h be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some h = On ± 1 can be determined constructively: for h = 5 there are four such determinants; for h = 7, six; for A = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one cl...