On two conjectures on real quadratic fields (Q1092110)

The authors use a result of T. Tatuzawa to prove that (with the possible exception of one value the following conjectures are true, i.e. that one of them is true and the other is true with the possible exception of one value): (C1) Let \(\ell\) be a square-free integer of the form \(\ell =q^ 2+4\); (q\(\in {\mathbb N})\). Then there exist just 6 quadratic fields \({\mathbb Q}(\sqrt{\ell})\) of class number one. (C2) Let \(\ell\) be a square-free integer of the form \(\ell =4q^ 2+1\) \((q\in {\mathbb N})\). Then there exist just 6 quadratic fields \({\mathbb Q}(\sqrt{\ell})\) of class number one. The authors acknowledge that this reviewer and \textit{H. C. Williams} have solved both conjectures using the generalized Riemann hypothesis. In point of fact there is a more general Mollin-Williams result for all real quadratic fields of Richaud-Degert type [see Théorie des nombres, C. R. Conf. Int., Québec 1987, 654--663 (1989; Zbl 0695.12002)].