arXiv2007
We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $α$, $β$, $γ$, $δ$, $ε$, and $ζ$. Particularly, numbers belong to Class $α$ are defined as $α=1+6 n$, and those of Class $β$, as $β=5+6n$, where $n=0,\pm1,\pm2,\pm3,\pm4,...$ These two Classes $α$ and $β$,contain: i) all prime numbers, except + 2, -2 and $\pm$3, which belong to $ε$, $δ$, and $γ$ Classes, respectively, and ii) all the other odd numbers, except those that are multiple of $\pm$3, according to the sequence $\pm$9, $\pm$15, $\pm$21, $\pm$27, ... Besides, products between numbers of the Class $α$, and also those between numbers of the Class $β$, generates numbers belonging to the Class $α$. On the other side, products between numbers of Class $α$ with numbers of Class $β$, result in numbers of Class $β$. Then, both Classes $α$ and $β$ include: i) all the prime numbers except $\pm$2 and $\pm$3, and ii) all the products between $α$ numbers, as $α\cdotα^{\prime}$; all the products between $β$ numbers, as $β\cdotβ^{\prime}$; and also all the products between numbers of Classes $α$ and $β$, as $α\cdotβ$, which necessarily are composite numbers, whose factorization is completely determined.