Definition of the invariant and the relationship with the compounds numbers. Generalisation of the Euler theorem

TL;DR

This paper introduces the concept of invariance related to numbers, generalizes Euler's theorem using these properties, and explores implications for primality testing and Carmichael numbers.

Contribution

It proposes a new invariant-based approach to generalize Euler's theorem and offers insights into primality testing and the nature of Carmichael numbers.

Findings

Generalization of Euler's theorem for all a^{φ(m)} values

Introduction of invariance properties for primality checks

Formulation of a hypothesis replacing the Totient function

Abstract

The purpose of this article is to introduce the concept of invariance and its properties. These properties can be used to check the primality of a number. Combining these properties with the Euler theorem, it is possible to generalize this theorem for all the values of $a^{\varphi(m)}$ where $0 < a < m {\pmod {m}}$ independently if a is co prime or not with m. As $a^{\varphi(m)+1} \equiv a$ if $m = a \cdot b$ and $GCD(a, b) = 1$. As the following steps, a new hypothesis is formulated regarding the substitution of the Totien function for an equivalent function that explains the Carmichael numbers. Keywords: Prime Numbers, Compound Numbers, Primality test, Euler theorem