Explicit representation of fundamental units of some quadratic fields
It is well-known that for any given and fixed square-free positive integer \(D\) the fundamental unit of the real quadratic field \(\mathbb{Q} (\sqrt {D})\) is explicitly described by using the continued fraction theory. In this paper, the author intends to provide an explicit unified form of the fundamental units of some real quadratic fields. Namely, for any square-free positive integer \(d\) congruent to \(1\bmod 4\), denote by \(k_d\), \(\varepsilon_d= (T_d+ U_d \sqrt {d})> 1\) the period in the continued fraction expansion of \(\omega_d= (1+ \sqrt {D}) /2\), the fundamental unit of \(\mathbb{Q} (\sqrt {D})\) respectively and put \(d= a^2+ b\) \((0< b\leq 2a)\). Then, under the assumption of \(k_d =3\), the author proves the following: in the case of even \(a\), the continued fraction expansion of \(\omega_d\) is \([a/2, \overline {1, 1, a-1} ]\), and moreover \((T_d, U_d)= (2a,2)\) and \(d= a^2+ 1\) hold. In the case of odd \(a\), the continued fraction expansion of \(\omega_d\) is \([(a+ 1)/2, \overline {l, l, a}]\) for a positive integer \(l\), and moreover \((T_d, U_d)= ((l^2+ 1)^2 r+l ((l^2+ 3), l^2+1)\) and \(d= (l^2+1)^2 r^2+ 2l (l^2+ 3)r+ l^2+4\) hold for a positive integer \(r\) satisfying \(a= (l^2+ 1)r+1\).
scientific article; zbMATH DE number 799701