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

Abstract

Suppose that $m\equiv 1\mathrm{mod}4$ is a prime and that $n\equiv 3\mathrm{mod}4$ 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 $m\equiv 3\mathrm{mod}4$, 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 $n+1$.

Final version

Cited by in corpus (1)

• On the sum of digits of $1/M$ in ${\mathbb{F}}_q[x]$