On Gauss’s class number problems

Abstract:

Let $h$ be the class number of binary quadratic forms (in Gauss’s formulation). All negative determinants having some $h = 6n \pm 1$ can be determined constructively: for $h = 5$ there are four such determinants; for $h = 7$, six; for $h = 11$, 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 class/genus for negative determinants is imbedded in the larger question of the existence of a determinant having a specific Abelian group as its composition group. All Abelian groups of order $< 25$ so exist, but the noncyclic groups of order 25, 49, and 121 do not occur. Positive determinants are treated by the same composition method. Although most positive primes of the form ${n^2} - 8$have $h = 1$, an interesting subset does not. A positive determinant of an odd exponent of irregularity also appears in the investigation. Gauss indicated that he could not find one.