Class number two problem for real quadratic fields with fundamental units with the positive norm (Q1285859)

For a square-free positive integer \(D\), let \(h_D\) and \(\varepsilon_D\) be the class number and the fundamental unit \((>1)\) of the real quadratic field \(\mathbb{Q} (\sqrt{D})\), respectively. Here \(\varepsilon_D\) can be expressed uniquely as \(\varepsilon_D= (t+u \sqrt{D})/2\), \(t\) and \(u\) being integers. In the papers [Proc. Japan Acad., Ser. A 67, 99-100 (1991; Zbl 0745.11050) and Math. Jap. 37, 1105-1115 (1992; Zbl 0771.11041)], \textit{S. G. Katayama} determined all \(D\)'s, with one possible exception of \(D\), satisfying \(h_D=2\), \(N\varepsilon_D=-1\) and \(1\leq u\leq 200\), where \(N\) denotes the norm map from \(\mathbb{Q} (\sqrt{D})\) to \(\mathbb{Q}\). In the paper under review, the authors consider an analogous class number problem in the case where \(N\varepsilon_D=1\), and determine all \(D\)'s with one possible exception such that \(h_D=2\), \(N\varepsilon_D=1\) and \(1\leq u\leq 100\), by using results of \textit{T. Tatuzawa} [Jap. J. Math. 21, 163-178 (1951; Zbl 0054.02302)] and \textit{H. Yokoi} [Nagoya Math. J. 132, 175-197 (1993; Zbl 0788.11047)], and further, by help of Y. Kida's UBASIC 86. The number of such \(D\)'s is exactly 125. The list of all such \(D\)'s is also given in a table.