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On certain real quadratic fields with class number one. (English) Zbl 0963.11064

For a positive square-free integer \(D\), denote by \(\varepsilon_D= (t+u\sqrt{D})/2\) the fundamental unit \((>1)\) of the real quadratic field \(k= \mathbb Q(\sqrt{D})\). Then, the reviewer showed that there exist exactly 30 real quadratic fields \(\mathbb Q(\sqrt{p})\) of class number one satisfying \(\varepsilon_p< 2p\) with one more possible exception of prime discriminant \(p\) [Nagoya Math. J. 120, 51-59 (1990; Zbl 0701.11046)]. Since then, Katayama, Mollin-Williams and others have shown some analogous results on the class number one problem, and in this paper the authors found the following five real quadratic fields \(\mathbb Q(\sqrt{D})\) with class number one for \(1\leq u\leq 100\) with one possible exception by the help of a computer and using Kida’s UBASIC 86: \((u,D)= (40,57), (77,893), (78,19), (84,22), (85,1397)\).

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R27 Units and factorization

Software:

UBASIC
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References:

[1] Karaali, F., and \.Işcan, H.: Class number two problem for real quadratic fields with fundamental units with the positive norm. Proc. Japan Acad., 74A , 139-141 (1998). · Zbl 0940.11052
[2] Katayama, S. I., and Katayama, S.-G.:[3] A note on the problem of Yokoi. Proc. Japan Acad., 67A , 26-28 (1991). · Zbl 0736.11065
[3] Kida, Y.: UBASIC 86. Nihonhyoronsha, Tokyo (1988). · Zbl 0692.10002
[4] Mollin, R. A., and Williams, H. C.: Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception). Number Theory (ed. Mollin, R. A.). Walter de Gruyter, Berlin-New York, pp. 417-425 (1990). · Zbl 0696.12004
[5] Mollin, R. A., and Williams, H. C.: A complete generalization of Yokoi’s \(p\)-invariants. Colloq. Math., 63 , fasc. 2, 285-294 (1992). · Zbl 0762.11035
[6] Yokoi, H.: The fundamental unit and class number one problem of real quadratic fields with prime discriminant. Nagoya Math. J., 120 , 51-59 (1990). · Zbl 0715.11057
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