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

DOI:
https://doi.org/10.1090/S0025-5718-1992-1136224-1

MathSciNet review:
1136224

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Abstract | References | Similar Articles | Additional Information

Abstract: It is the primary purpose of the paper to determine all real quadratic fields of class number when (with one possible exception). Here, *k* is the period length of the continued fraction expansion of either , in the case or 3 , or of , in the case .

**[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 (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 95–101. MR**1151858****[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*, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 654–663. MR**1024594****[4]**R. A. Mollin and H. C. Williams,*Class number one for real quadratic fields, continued fractions and reduced ideals*, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 481–496. MR**1123091****[5]**R. A. Mollin and H. C. Williams,*Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception)*, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 417–425. MR**1106676****[6]**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), no. 1, 20–25. MR**1103974****[7]**A. J. Stephens and H. C. Williams,*Some computational results on a problem concerning powerful numbers*, Math. Comp.**50**(1988), no. 182, 619–632. MR**929558**, https://doi.org/10.1090/S0025-5718-1988-0929558-3**[8]**Tikao Tatuzawa,*On a theorem of Siegel*, Jap. J. Math.**21**(1951), 163–178 (1952). MR**0051262**, https://doi.org/10.4099/jjm1924.21.0_163**[9]**Hisao Taya and Nobuhiro Terai,*Determination of certain real quadratic fields with class number two*, Proc. Japan Acad. Ser. A Math. Sci.**67**(1991), no. 5, 139–144. MR**1114957****[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), no. 177, 405–423. MR**866124**, https://doi.org/10.1090/S0025-5718-1987-0866124-1

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1136224-1

Keywords:
Real quadratic field,
class number,
continued fraction

Article copyright:
© Copyright 1992
American Mathematical Society