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Some periodic continued fraction expansions and fundamental units of quadratic orders. (Einige periodische Kettenbruchentwicklungen und Grundeinheiten quadratischer Ordnungen.) (German) Zbl 0764.11009
Let $$D$$ be a discriminant of a quadratic number field and let $$\omega_ D=(1+\sqrt{D})/2$$ if $$D\equiv 1\pmod 4$$ and $$\omega_ D=\sqrt{D}/2$$ otherwise. The author computes the period length of the continued fraction of $$\omega_ D$$ and the fundamental unit of $$\mathbb{Z}[\omega_ D]$$ in the case when $$D=(\ell p^ k+\lambda c)^ 2+4\mu p^ k q$$, where $$k\geq 2$$, $$q\geq 1$$, $$\ell$$, $$q$$ are integers, $$\lambda,\mu=\pm1$$, $$p=c\ell+\lambda\mu\geq 2$$ and $$q$$ divides $$c$$. This covers several special cases considered earlier by several other authors [L. Bernstein, J. Number Theory 8, 446–491 (1976; Zbl 0352.10002); M. D. Hendy, Math. Comput. 28, 267–277 (1974; Zbl 0275.12007); C. Levesque, J. Math. Phys. Sci. 22, No. 1, 11–44 (1988; Zbl 0645.10010); C. Levesque and G. Rhin, Util. Math. 30, 79–107 (1986; Zbl 0615.10014); M. Nyberg, Norsk Mat. Tidsskr. 31, 95–99 (1949; Zbl 0040.30706)].

##### MSC:
 11A55 Continued fractions 11R11 Quadratic extensions 11R27 Units and factorization
##### Keywords:
period length; continued fraction; fundamental unit
Full Text:
##### References:
 [1] T. Azuhata, On the Fundamental Units and the Class Numbers of Real Quadratic Fields II, Tokyo J. Math.10 (1987), 259–270. · Zbl 0659.12008 [2] L. Bernstein, Fundamental Units and Cycles in the Period of Real Quadratic Number Fields I, II. J. Number Theory8 (1976), 446–491; Pacific J. Math.63 (1976), 37-61 and 63-78. · Zbl 0352.10002 [3] L. E. Dickson, History of the Theory of Numbers, vol. II. Chelsea 1971. [4] M. D. Hendy, Applications of a Continued Fraction Algorithm to Some Class Number Problems. Math. Comp.28 (1974), 267–277. · Zbl 0275.12007 [5] C. Levesque, Continued Fraction Expansions and Fundamental Units. J. Math. Phys. Sci.22 (1988), 11–44. · Zbl 0645.10010 [6] C. Levesque andG. Rhin, A few classes of periodic continued fractions. Utilitas Math.30 (1986), 79–107. · Zbl 0615.10014 [7] M. Nyberg, Culminating and almost culminating continued fractions (in Norwegian). Norsk Mat. Tidsskr.31 (1949), 95–99. [8] O. Perron, Die Lehre von den Kettenbrüchen, Bd. 1. Teubner 1954. · Zbl 0056.05901 [9] H. C. Williams, A note on the period length of the continued fraction expansion of certain {ie-1}, Utilitas Math.28 (1985), 201–209. · Zbl 0586.10004 [10] Y. Yamamoto, Real quadratic number fields with large fundamental units. Osaka J. Math.8 (1971), 261–270. · Zbl 0243.12001
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