Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On real quadratic fields of class number two

Authors: R. A. Mollin and H. C. Williams
Journal: Math. Comp. 59 (1992), 625-632
MSC: Primary 11R11; Secondary 11R29
MathSciNet review: 1136224
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is the primary purpose of the paper to determine all real quadratic fields $ Q(\sqrt d )$ of class number $ h(d) = 2$ when $ k \leq 24$ (with one possible exception). Here, k is the period length of the continued fraction expansion of either $ \omega = \sqrt d $, in the case $ d \equiv 2$ or 3 $ \pmod 4$, or of $ \omega = (1 + \sqrt d )/2$, in the case $ d \equiv 1\, \pmod 4$.

References [Enhancements On Off] (What's this?)

  • [1] R. A. Mollin and H. C. Williams, On a solution of a class number two problem for a family of real quadratic fields, Computational Number Theory (A. Pethö, M. Pohst, H. Williams, and H. Zimmer, eds.), Walter de Gruyter, Berlin, 1991, pp. 95-101. MR 1151858 (93d:11118)
  • [2] -, Computation of the class numbers of a real quadratic field, Advances in the Theory of Computing and Comput. Math. (to appear).
  • [3] R. A. Mollin and H. C. Williams, Prime-producing quadratic polynomials and real quadratic fields of class number one, Number Theory (J. M. DeKoninck and C. Levesque, eds.), Walter de Gruyter, Berlin, 1989, pp. 654-663. MR 1024594 (90m:11153)
  • [4] -, Class number one for real quadratic fields, continued fractions and reduced ideals, Number Theory and Applications (R. A. Mollin, ed.), Kluwer, Dordrecht, 1989, pp. 481-496. MR 1123091 (92f:11143)
  • [5] -, Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception), Number Theory (R. A. Mollin, ed.), Walter de Gruyter, Berlin, 1990, pp. 417-425. MR 1106676 (92f:11144)
  • [6] -, 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)
  • [7] A. J. Stephens and H. C. Williams, Some computational results on a problem concerning powerful numbers, Math. Comp. 50 (1988), 619-632. MR 929558 (89d:11091)
  • [8] T. Tatuzawa, On a theorem of Siegel, Japan J. Math. 21 (1951), 163-178. MR 0051262 (14:452c)
  • [9] H. Taya and N. Terai, Determination of certain real quadratic fields with class number two, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 139-144. MR 1114957 (92h:11091)
  • [10] 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)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11R11, 11R29

Retrieve articles in all journals with MSC: 11R11, 11R29

Additional Information

Keywords: Real quadratic field, class number, continued fraction
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society