Quadratic residue covers for certain real quadratic fields
Authors:
R. A. Mollin and H. C. Williams
Journal:
Math. Comp. 62 (1994), 885-897
MSC:
Primary 11R11; Secondary 11Y40
DOI:
https://doi.org/10.1090/S0025-5718-1994-1218346-1
MathSciNet review:
1218346
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let with
and
. If
is a finite set of primes such that for each
there exists some
for which the Legendre symbol
, we call
a quadratic residue cover (QRC) for the quadratic fields
. It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of
when
is the discriminant of
. Also, QRCs are computed for all
.
- [1] H. Cohn, A second course in number theory, Wiley, New York, 1962. MR 0133281 (24:A3115)
- [2] F. Halter-Koch, Einige periodische Kettenbruchentwicklungen und Grundeinheiten quadratischer Ordnungen, Abh. Math. Sem. Univ. Hamburg 59 (1989), 157-169. MR 1049893 (91h:11115)
- [3] H. K. Kim, M.-G. Leu, and T. Ono, On two conjectures on real quadratic fields, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 222-224. MR 907000 (88k:11072)
- [4] D. H. Lehmer, The lattice points of an n-dimensional tetrahedron, Duke Math. J. 7 (1940), 341-353. MR 0003013 (2:149g)
- [5] S. Louboutin, R. A. Mollin, and H. C. Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, prime producing quadratic polynomials and quadratic covers, Canad. J. Math. 44 (1992), 824-842. MR 1178571 (93h:11117)
- [6] R. A. Mollin, Powers in continued fractions and class numbers of real quadratic fields, Utilitas Math. 42 (1992), 25-30. MR 1199085 (93k:11102)
- [7] R. A. Mollin and H. C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period length less than 25, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 20-25. MR 1103974 (92c:11113)
- [8] -, Affirmative solution of a conjecture related to a sequence of Shanks, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 70-72. MR 1105525 (92c:11122)
- [9] A. S. Pen and B. F. Skubenko, Estimation from above of the period of a quadratic irrationality, Math. Notes Acad. Sci. USSR 5 (1969), 247-250. MR 0245524 (39:6830)
- [10] B. Rosser, On the first case of Fermat's last theorem, Bull. Amer. Math. Soc. 45 (1939), 636-640. MR 0000025 (1:5b)
- [11] D. Shanks, On Gauss's class number problems, Math. Comp. 23 (1969), 151-163. MR 0262204 (41:6814)
- [12] T. Tatuzawa, On a theorem of Siegel, Japan J. Math. 21 (1951), 163-178. MR 0051262 (14:452c)
- [13] H. C. Williams and M. C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 48 (1987), 405-423. MR 866124 (88i:11099)
Retrieve articles in Mathematics of Computation with MSC: 11R11, 11Y40
Retrieve articles in all journals with MSC: 11R11, 11Y40
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1994-1218346-1
Article copyright:
© Copyright 1994
American Mathematical Society