On Gauss’s class number problems

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Abstract
Let h h be the class number of binary quadratic forms (in Gauss’s formulation). All negative determinants having some h = 6 n ± 1 h = 6n \pm 1 can be determined constructively: for h = 5 h = 5 there are four such determinants; for h = 7 h = 7 , six; for h = 11 h = 11 , four; and for h = 13 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 > 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 {n^2} - 8 have h = 1 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.

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