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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On Gauss's class number problems

Author: Daniel Shanks
Journal: Math. Comp. 23 (1969), 151-163
MSC: Primary 10.66
MathSciNet review: 0262204
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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.

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Article copyright: © Copyright 1969 American Mathematical Society