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On two conjectures on real quadratic fields. (English) Zbl 0624.12002

The authors use a result of T. Tatuzawa to prove that (with the possible exception of one value the following conjectures are true, i.e. that one of them is true and the other is true with the possible exception of one value):
(C1) Let \(\ell\) be a square-free integer of the form \(\ell =q^ 2+4\); (q\(\in {\mathbb N})\). Then there exist just 6 quadratic fields \({\mathbb Q}(\sqrt{\ell})\) of class number one.
(C2) Let \(\ell\) be a square-free integer of the form \(\ell =4q^ 2+1\) \((q\in {\mathbb N})\). Then there exist just 6 quadratic fields \({\mathbb Q}(\sqrt{\ell})\) of class number one.
The authors acknowledge that this reviewer and H. C. Williams have solved both conjectures using the generalized Riemann hypothesis. In point of fact there is a more general Mollin-Williams result for all real quadratic fields of Richaud-Degert type [see Théorie des nombres, C. R. Conf. Int., Québec 1987, 654–663 (1989; Zbl 0695.12002)].
Reviewer: R.A.Mollin

MSC:

11R11 Quadratic extensions
11R23 Iwasawa theory

Citations:

Zbl 0695.12002
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References:

[1] T. Tatuzawa: On a theorem of Siegel. Japan. J. Math., 21, 163-178 (1951). · Zbl 0054.02302
[2] H. Yokoi: Class-number one problem for certain kind of real quadratic fields. Proc. International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields, June 24-28, 1986, Katata, Japan, pp. 125-137. · Zbl 0612.12010
[3] R. Sasaki: A characterization of certain real quadratic fields. Proc. Japan Acad., 62A, 97-100 (1986). · Zbl 0593.12002 · doi:10.3792/pjaa.62.97
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