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 $m$ . We obtain a relation between the class number of the imaginary quadratic field $ℚ (\sqrt{- n m})$ 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 $n + 1$ .

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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

Authors

Kalyan Chakraborty
Kerala School of Mathematics KCSTE Kerala India

Krishnarjun Krishnamoorthy
Harish-Chandra Research Institute Prayagraj India

Vol. 319, No. 1, 2022

Kalyan Chakraborty and Krishnarjun Krishnamoorthy

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors