On some symmetries of the base $ n $ expansion of $ 1/m $: the class number connection

Kalyan Chakraborty and Krishnarjun Krishnamoorthy

Vol. 319 (2022), No. 1, 39–53

DOI: 10.2140/pjm.2022.319.39

Abstract

Suppose that $m\equiv 1\mathrm{mod}<!--FUNCTION\; APPLICATION-->4$ is a prime and that $n\equiv 3\mathrm{mod}<!--FUNCTION\; APPLICATION-->4$ 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 $1∕m$.

Secondly, if $m\equiv 3\mathrm{mod}<!--FUNCTION\; APPLICATION-->4$, we study some convoluted sums involving the base $n$ digits of $1∕m$ and arrive at certain congruence relations involving the class number of $\mathbb{Q}(\sqrt{-m})$ modulo certain primes $p$ which properly divide $n+1$.

Keywords

class numbers, 3-divisibility, quadratic fields, base $n$ representation.

Mathematical Subject Classification

Primary: 11A07, 11R29

Milestones

Received: 17 October 2021

Revised: 22 February 2022

Accepted: 26 March 2022

Published: 28 August 2022