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

TL;DR

This paper explores the relationship between the base n expansion of 1/m and class numbers of certain quadratic fields, revealing new connections and congruences involving prime moduli and digit patterns.

Contribution

It establishes a novel link between the digits of 1/m in base n and the class numbers of imaginary quadratic fields, extending previous number theory results.

Findings

Relation between class number of Q(√-nm) and base n digits of 1/m

Congruence relations involving class number of Q(√-m) and primes dividing n+1

New insights into digit symmetries in base n expansions of 1/m

Abstract

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 $.