Suppose that
$m\equiv 1\phantom{\rule{0.3em}{0ex}}\mathrm{mod}\phantom{\rule{0.3em}{0ex}}4$ is a
prime and that
$n\equiv 3\phantom{\rule{0.3em}{0ex}}\mathrm{mod}\phantom{\rule{0.3em}{0ex}}4$ is a
primitive root modulo
$m$.
We obtain a relation between the class number of the imaginary quadratic field
$\mathbb{Q}(\sqrt{nm})$ and the digits
of the base
$n$
expansion of
$1\u2215m$.
Secondly, if
$m\equiv 3\phantom{\rule{0.3em}{0ex}}\mathrm{mod}\phantom{\rule{0.3em}{0ex}}4$,
we study some convoluted sums involving the base
$n$ digits
of
$1\u2215m$
and arrive at certain congruence relations involving the class number of
$\mathbb{Q}(\sqrt{m})$ modulo certain
primes
$p$ which
properly divide
$n+1$.
