On continued fractions of real quadratic fields with period six (Q2892549)

Let \(D = a^2 + b\) be an integer where \(a\) and \(b\) are integers such that \(0<b \leq 2a\), \(\omega_D = \sqrt{D}\) be such that its continued fraction expansion is of period \(l(\omega_{D}) = 6\) and \(k = \mathbb Q(\sqrt{D})\). In this paper, the authors determine the general forms of the continued fraction expansions of \(\omega_{D}\) in the two casesNEWLINENEWLINE (i) \(D\equiv 2 \bmod 4\), \(a\equiv 1 \bmod 4 \) and \(b\equiv 1\bmod 4\),NEWLINENEWLINE(ii) \(D\equiv 3 \bmod 4\), \(a\equiv 0 \bmod 4 \) and \(b\equiv 3\bmod 4\).NEWLINENEWLINEFurthermore, they obtain some results on Yokoi's invariant \([\frac{ t_{D}}{ u_{D}^{2}}]\); where \([x]\) means the greatest integer not greater than \(x\) and \( t_{D} + u_{D}\sqrt{D}\) is the fundamental unit of \(k\). They also give some numerical examples which are a revision of the tables of the work of \textit{A. Pekin} and \textit{H. Iscan} [``Continued fractions of period six and explicit representations of fundamental units of some real quadratic fields'', J. Indian Math. Soc., New Ser. 72, No. 1--4, 183--194 (2005; Zbl 1186.11006)].