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A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin ,
L.-C. Zhang  and
Paula Kemp
R. A. Mollin
Affiliation:
Mathematics Department University of Calgary Calgary, Alberta T2N 1N4 e-mail:ramollin@acs.ucalgary.ca
L.-C. Zhang
Affiliation:
Mathematics Department Southwest Missouri State University Springfield, Missouri 65804 U.S.A. e-mail:liz917f@smsvma.bitnet
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Abstract

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In this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

Keywords

11R1111R0911R29class numberprimitive idealdivisor functionLagrange neighbourERD-typequadratic fields
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 1 , 01 March 1994 , pp. 90 - 96
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. F., Halter-Koch, Quadratische Ordnungen mit grosser Klassenzahl, J. Number Theory 34(1990), 82–94.Google Scholar
2. F., Halter-Koch, Quadratische Ordnungen mit grosser Klassenzahl, II, J. Number Theory 44(1993), 166–171.Google Scholar
3. Mollin, R. A., On the divisor function and class numbers of real quadratic fields I, Proc. Japan Acad. Ser. A 6(1990), 109–111.Google Scholar
4. Mollin, A., On the insolubility of a class of diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richard-Degert type, Nagoya Math. J. 105( 1987), 39–47.Google Scholar
5. Mollin, A., Class number one criteria for real quadratic fields II, Proc. Japan Acad. 63(1987), 162–164.Google Scholar
6. Mollin, R. A. and Williams, H. C., Class Number Problems for real Quadratic Fields, Number Theory and Cryptography, (ed. J. H. Loxton), London Lecture Note series 154, 1990, 77–105.Google Scholar
7. Mollin, R. A. and Williams, H. C., On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. 112 (1988), 143–151.Google Scholar
8. Mollin, R. A. and Williams, H. C., Solution of the class number one problem for real quadratic fields of extended Richard-Degert type (with one possible exception), Number Theory (ed. R. A. Mollin), Walter de Gruyter, Berlin-New York, 1990,417–425.Google Scholar
9. Mollin, R. A. and Williams, H. C., On a determination of real quadratic fields of class number one and related continued fraction period length less than25, Proc. Japan Acad. Ser. A 67(1991), 20–25.Google Scholar
10. Mollin, R. A. and Williams, H. C., On real quadratic fields of class number two, Math. Comp. 59(1992), 625–632.Google Scholar
11. Mollin, R. A. and Zhang, L.-C., Reduced ideals, the divisor function, continued fractions and class numbers of real quadratic fields, Publ. Math. Debrecen, to appear.Google Scholar
12. Williams, H. C. and M. C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 177(1987), 405–423.Google Scholar