  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
