Vol. 319, No. 1, 2022

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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
Abstract

Suppose that m 1mod4 is a prime and that n 3mod4 is a primitive root modulo m. We obtain a relation between the class number of the imaginary quadratic field (nm) and the digits of the base n expansion of 1m.

Secondly, if m 3mod4, we study some convoluted sums involving the base n digits of 1m and arrive at certain congruence relations involving the class number of (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