Class number one problem for real quadratic fields of a certain type (Q2880461)

This paper tackles an old but still appealing problem of the determination of real quadratic fields \(\mathbb Q(\sqrt{d})\) of discriminant \(d\) of R-D type (i.e., \(d=a^2n^2\pm ka\), where \(k=1, 2\) or \(4\)) with class number one. We do know using a theorem of Siegel that the number of such fields is finite. However, Siegel's result used here being ineffective, an enumeration of these fields has not been possible, except in the case of \(a=1\) [\textit{A. Biró}, Acta Arith. 107, No. 2, 179--194 (2003; Zbl 1154.11339); Acta Arith. 106, No. 1, 85--104 (2003; Zbl 1154.11338); \textit{J. Lee}, Acta Arith. 140, No. 1, 1--29 (2009; Zbl 1241.11124); \textit{D. Byeon} et al., Acta Arith. 126, No. 2, 99--114 (2007; Zbl 1125.11059)]. The current paper studies a particular case when \(d=(an)^2+4a\), where \(a, n\) are odd positive integers such that \(43\cdot 181\cdot\cdot 353\) divides \(n\) and it is shown that such discriminants have class number greater than one. Following the method in [\textit{A. Biró} and \textit{A. Granville}, J. Number Theory 132, No. 8, 1807--1829 (2012; Zbl 1276.11180)], for the specific \(d\) given above, using a real character modulo \(q\) (with \(q|n\)) and the fact that the class number \(h(d)=1\), the following identity is proved: NEWLINE\[NEWLINEqh(-q)h(-qd)= n\left(a+\left(\frac{a}{q}\right)\right)\frac{1}{6} \prod_{p|q} (p^2-1),NEWLINE\]NEWLINE which, along with considerations such as the powers of \(2\) dividing each side of the identity, allows the author to conclude the proof.