On S. Chowla's conjecture (Q1912694)

In connection with the class number one problem for real quadratic fields started from S. Chowla's conjecture [cf. \textit{S. Chowla}, and \textit{J. Friedlander}, Glasg. Math. J. 17, 47-52 (1976; Zbl 0323.12006)], the authors intend to provide a necessary condition for real quadratic fields with certain type to have class number one. Namely, by using the Dedekind sum, the Hirzebruch sum on the simple continued fraction development of a real quadratic irrational number and Kubota's class number formula for biquadratic fields, they prove the following: If the real quadratic field \(\mathbb{Q}(\sqrt p)\) has class number one for a prime \(p= 4q^2+ 1\) with a prime \(q\equiv 3\pmod 4\) and \(q> 3\), then the biquadratic field \(K= \mathbb{Q}(\sqrt{- q}, \sqrt p)\) has class number \(h_K= (p^2- 1)/6\).