A Generalization of A Result of Gauss on Primitive Root

TL;DR

This paper generalizes Gauss's classical results on primitive roots modulo primes, extending the congruences for sums and products of primitive roots to a broader class of integers with the same index.

Contribution

It introduces generalized congruences for sums and products of integers with the same index modulo any integer, extending Gauss's primitive root results.

Findings

Generalized congruences for sums of integers with the same index

Generalized congruences for products of integers with the same index

Extension of Gauss's primitive root results to composite moduli

Abstract

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $\mu(p-1)$ modulo $p$, where $\mu$ is the M\"{o}bius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.