On certain real quadratic fields with class number one (Q1594782)

For a positive square-free integer \(D\), denote by \(\varepsilon_D= (t+u\sqrt{D})/2\) the fundamental unit \((>1)\) of the real quadratic field \(k= \mathbb Q(\sqrt{D})\). Then, the reviewer showed that there exist exactly 30 real quadratic fields \(\mathbb Q(\sqrt{p})\) of class number one satisfying \(\varepsilon_p< 2p\) with one more possible exception of prime discriminant \(p\) [Nagoya Math. J. 120, 51-59 (1990; Zbl 0701.11046)]. Since then, Katayama, Mollin-Williams and others have shown some analogous results on the class number one problem, and in this paper the authors found the following five real quadratic fields \(\mathbb Q(\sqrt{D})\) with class number one for \(1\leq u\leq 100\) with one possible exception by the help of a computer and using Kida's UBASIC 86: \((u,D)= (40,57), (77,893), (78,19), (84,22), (85,1397)\).