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On determining certain real quadratic fields with class number one and relating this property to continued fractions and primality properties

Published online by Cambridge University Press:  22 January 2016

Eugène Dubois  and
Claude Levesque
Eugène Dubois
Affiliation:
Département de MathématiquesISMRA & UNIVERSITE 14,032 CAEN-FRANCE
Claude Levesque
Affiliation:
Département de Mathématiques Université LAVALQUEBEC CANADA G1K 7P4
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Thanks to K. Heegner [He], A. Baker [Ba] and H. Stark [S], we know that there are nine imaginary quadratic fields of class number one. Gauss conjectured that there are infinitely many real quadratic fields of class number one, but the conjecture is still open.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 124 , December 1991 , pp. 157 - 180
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[Ba] Baker, A., Linear forms in the logarithms of algebraic numbers, Mathematica, 13 (1966), 204–216.Google Scholar
[Bel] Bernstein, L., Fundamental units and cycles in the period of real quadratic number fields, I, Pacific J. Math., 63, N°l, (1976), 37–61; J. Number Theory, 8 (1976), 446–491.CrossRefGoogle Scholar
[Be2] Bernstein, L., Fundamental units and cycles in the period of real quadratic number fields, II, Pacific J. Math., 63, N°2, (1976), 63–78.CrossRefGoogle Scholar
[H-K] Halter-Koch, F., Einige periodische Kettenbruchentwicklungen und Grundein-heiten quadratischer ordnungen, Abh. Math. Sem. Univ. Hamburg, 59 (1989), 157–169.Google Scholar
[He] Heegner, K., Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.CrossRefGoogle Scholar
[Ho] Hoffstein, J., On the Siegel-Tatuzawa theorem, Acta Arith., 38 (1980), 167–174.CrossRefGoogle Scholar
[Le] Levesque, C, Continued fraction expansions and fundamental units, J. Math. Phy. Sci., 22, N°l, (1988), 11–44.Google Scholar
[Le-R] Levesque, C. and Rhin, G., A few classes of periodic continued fractions, Utilitas Mathematica, 30 (1986), 79–107.Google Scholar
[Lo1] Louboutin, S., Arithmétique des corps quadratiques réels et fractions continues, Thèse de doctorat, Univ. Paris 7, Juin 1987.Google Scholar
[Lo2] Louboutin, S., Continued fractions and real quadratic fields, J. Number Theory, 30, N°2, (1988), 167–176.CrossRefGoogle Scholar
[Lu] Lu, H., On the class number of real quadratic fields, Scientia Sinica, 2 (1979), 118–130.Google Scholar
[M1] Mollin, R. A., On the insolubility of a class of diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type, Nagoya Math. J., 105 (1987), 39–47.CrossRefGoogle Scholar
[M2] Mollin, R. A., On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J., 112 (1988).CrossRefGoogle Scholar
[M3] Mollin, R. A., Class numbers of quadratic fields determined by solvability of diophan tine equations, Math. Comp., 48, N° 177 (1987), 233–242.CrossRefGoogle Scholar
[M-W] 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), from Number Theory, Proceedings (1990) edited by Mollin, R. A., published by W de G. CrossRefGoogle Scholar
[N] Nyberg, M., Culminating and almost culminating continued fractions, Norsk. Mat. Tidsskr., 31 (1949), 95–99, (in Norwegian), MR A56–16.Google Scholar
[S] Stark, H. M., A complete determination of the complex quadratic fields of class number one, Michigan Math. J., 14 (1967), 1–27.CrossRefGoogle Scholar
[T] Tatuzawa, T., On a theorem of Siegel, Japan J. Math., 21 (1951), 163–178.CrossRefGoogle Scholar
[W] Williams, H. C., A note on the period length of the continued fraction expansion of certain , Utilitas Mathematica, 28 (1985), 201–209.Google Scholar