A Generalization of Euler's Criterion to Composite Moduli

TL;DR

This paper extends Euler's Criterion to composite moduli by providing a necessary and sufficient condition for binomial congruence solvability without prime factorization, utilizing idempotent numbers and concepts of order and primitive roots.

Contribution

It introduces a generalized Euler's Criterion applicable to composite moduli, broadening the theoretical framework for solving binomial congruences.

Findings

Provides a necessary and sufficient condition for binomial congruence solvability

Circumvents prime factorization in the analysis

Utilizes idempotent numbers, order, and primitive roots

Abstract

A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.