On Gauss’s class number problems. (English) Zbl 0177.07103

Summary: 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.


11E41 Class numbers of quadratic and Hermitian forms
11E16 General binary quadratic forms
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