On the location and classification of all prime numbers

Abstract

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, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly, numbers belong to Class $\alpha$ are defined as $\alpha=1+6 n$, and those of Class $\beta$, as $\beta=5+6n$, where $n=0,\pm1,\pm2,\pm3,\pm4,...$ These two Classes $\alpha$ and $\beta$,contain: i) all prime numbers, except + 2, -2 and $\pm$3, which belong to $\epsilon$, $\delta$, and $\gamma$ 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 $\alpha$, and also those between numbers of the Class $\beta$, generates numbers belonging to the Class $\alpha$. On the other side, products between numbers of Class $\alpha$ with numbers of Class $\beta$, result in numbers of Class $\beta$. Then, both Classes $\alpha$ and $\beta$ include: i) all the prime numbers except $\pm$2 and $\pm$3, and ii) all the products between $\alpha$ numbers, as $\alpha\cdot\alpha^{\prime}$; all the products between $\beta$ numbers, as $\beta\cdot\beta^{\prime}$; and also all the products between numbers of Classes $\alpha$ and $\beta$, as $\alpha\cdot\beta$, which necessarily are composite numbers, whose factorization is completely determined.