Class number one criteria for real quadratic fields. II (Q579314)

In the previous paper (see the preceding review), the author considered real quadratic fields \({\mathbb{Q}}(\sqrt{n})\) with \(n\equiv 1\) (mod 4) under a certain assumption, and gave criteria for these real quadratic fields to have class number one. In this paper, he deals with positive square-free integers n of R-D type, and intends to generalize results by \textit{T. Azuhara} [Nagoya Math. J. 95, 123-135 (1984; Zbl 0533.12008)], \textit{R. Sasaki} [to appear in Nagoya Math. J., Zbl 0612.12003] and himself [J. Number Theory 24, 7-19 (1986; Zbl 0591.12006)]. Namely, he gives a necessary condition for real quadratic fields \({\mathbb{Q}}(\sqrt{n})\) such that \(n=m^ 2+r>7\), 4m\(\equiv 0\) (mod r), \(-m<r\leq m\) and \(m\not\equiv 1\) (mod 4) to have class number one.