On some symmetries of the base \( n \) expansion of \( 1/m \) : The Class Number connection

Suppose that \( m\equiv 1\mod 4 \) is a prime and that \( n\equiv 3\mod 4 \) is a primitive root modulo \( m \). In this paper we obtain a relation between the class number of the imaginary quadratic field \( \Q(\sqrt{-nm}) \) and the digits of the base \( n \) expansion of \( 1/m \). Secondly, if \( m\equiv 3\mod 4 \), we study some convoluted sums involving the base \( n \) digits of \( 1/m \) and arrive at certain congruence relations involving the class number of \( \Q(\sqrt{-m}) \) modulo certain primes \( p \) which properly divide \( n+1 \).