On some symmetries of the base \(n\) expansion of \(1/m\): the class number connection (Q2171870)

Let \(m \in \mathbb P\) be a prime and \(n\) a primitive root modulo \(m\). If \(m \equiv 3 \bmod 4\), \textit{K. Girstmair} [Am. Math. Mon. 101, No. 10, 997--1001 (1994; Zbl 0839.11049)] showed how to obtain the class number of \(\mathbb Q(\sqrt{-m})\) from the digits of the expansion of \(1/m\) in base \(n\). Theorem 3 of the paper under review gives a similar formula for the class number of \(\mathbb Q(\sqrt{-nm})\), supposing that \(m \equiv 1 \bmod 4\) and \(n \equiv 3 \bmod 4\). Both results can be derived from Dirichlet's class number formula with some more or less elementary calculations. Furthermore, the authors apply their method to obtain some special divisibility results for class numbers of quadratic number fields.