On Composite Moduli from the Viewpoint of Idempotent Numbers

TL;DR

This paper introduces idempotent numbers to study composite moduli without prime factorization, generalizing key theorems like Euler-Fermat and primitive roots, and extends conditions for binomial congruence solvability.

Contribution

It presents a new class of numbers, idempotent numbers, and generalizes fundamental theorems related to composite moduli, avoiding prime factorization complexities.

Findings

Introduction of idempotent numbers for composite moduli analysis

Generalizations of Euler-Fermat Theorem and primitive roots

Main result: generalized condition for binomial congruence solvability

Abstract

The purpose of this paper is to introduce basic concepts that are fundamental in the examination of composite moduli, while avoiding the notoriously difficult problem of prime-factorization. We introduce a new class of numbers, called idempotent numbers, that is unavoidable when researching composite moduli. Among many interesting results, we give generalizations of well-known theorems and definitions, such as the Euler-Fermat Theorem and the concept of primitive roots. We consider the generalization of the equivalence condition for the solvability of a binomial congruence to be the main result of our paper.