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

Shanks, Daniel

Math. Comput. 23, 151-163 (1969).

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.

MSC:

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

Keywords:

class number of binary quadratic forms; class numbers for determinants; class numbers for discriminants