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

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