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 be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some can be determined constructively: for there are four such determinants; for , six; for , four; and for , 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 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 have , 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.

**[1]**Carl Friedrich Gauss,*Disquisitiones arithmeticae*, Translated into English by Arthur A. Clarke, S. J, Yale University Press, New Haven, Conn.-London, 1966. MR**0197380****[2]**S. Chowla, ``Heilbronn's class-number theorem,''*Proc. Indian Acad. Sci. Sect. A*, v. 1, 1934, pp. 74-76.**[3]**C. F. Gauss,*Untersuchungen über höhere Arithmetik*, Chelsea, New York, 1965, (Reprint of Maser Translation). The English translation [1] is garbled here. MR**32**#5488.**[4]**Daniel Shanks,*On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎*, Math. Comp.**14**(1960), 320–332. MR**0120203**, 10.1090/S0025-5718-1960-0120203-6**[5]**L. E. Dickson,*Introduction to the Theory of Numbers*, Univ. of Chicago Press, Chicago, 1929, Theorem 85, p. 111.**[6]**K. E. Kloss,*Some number-theoretic calculations*, J. Res. Nat. Bur. Standards Sect. B**69B**(1965), 335–336. MR**0190057****[7]**H. J. Smith,*Report on the Theory of Numbers, Chelsea*, New York, 1964, reprint.**[8]**L. E. Dickson,*History of the Theory of Numbers*, Vol. 3, Stechert, New York, 1934, reprint, Chapter V.**[9]**Gordon Pall, ``Note on irregular determinants,''*J. London Math. Soc.*, v. 11, 1936, pp. 34-35.

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DOI:
https://doi.org/10.1090/S0025-5718-1969-0262204-1

Article copyright:
© Copyright 1969
American Mathematical Society