Explicit representation of fundamental units of some real quadratic fields. II (Q1355088)

As a continuation of his preceding paper [\textit{K. Tomita}, Proc. Japan Acad., Ser. A 71, 41-43 (1995; Zbl 0831.11058)], where he described explicitly the fundamental unit of a real quadratic field \(Q (\sqrt d)\) such that the period \(k_d\) of the continued fraction expansion of \(\omega_d= (1+ \sqrt d)/2\) is equal to 3, the author makes an analogous study in the case of \(k_d= 4\), 5. Namely, he provides, in this paper, the explicit forms of \(T_d\) and \(U_d\) in the fundamental unit \(\varepsilon_d=(T_d+ U_d \sqrt d)/2\) \((>1)\) of \(Q(\sqrt d)\) and \(d\) itself by using at most four parameters appearing in the continued fraction expansion of \(\omega_d\). For example, for a positive square-free integer \(d\) congruent to 1 modulo 4, if we assume that \(k_d=5\) and in \(d= a^2+ b\) \((0<b \leq 2a)\) \(a\) is even, then \[ \omega_d= [a/2, \overline {1,\ell, \ell,1,a -1}] \] for an integer \(\ell\geq 0\), and \[ T_d= A^2r+ B,\;U_d =A,\;d= A^2r^2 +2Br +C, \] where \(A= \ell^2+2 \ell+2\), \(B= (\ell^2+ \ell)A+ \ell^2,\) \(C= (\ell^2+3) \ell^2+ 2(\ell^2-1) \ell+1\) and \(r\) is a nonnegative integer determined uniquely by \(a= Ar+ \ell^2+\ell\).