# Decimal canonical combinatorial number matrix

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The decimal canonical combinatorial number matrix is a special arrangement of the natural numbers, used to develop new mathematical tools in number theory; sometimes it is refered to as the decimal number matrix. A submatrix is the prime canonical combinatorial number matrix with the same purpose.

## Definition

Let ${\displaystyle T = T_{ij}}$ be the decimal canonical combinatorial number matrix:

${\displaystyle T_{ij} = \left( \begin{array}{llll} a_{01} & a_{02} & \ldots & a_{010} \\ a_{11} & a_{12} & \ldots & a_{120} \\ a_{21} & a_{22} & \ldots & a_{230} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{n 10} \\ \end{array} \right) }$

with the following properties:

${\displaystyle i = 0:a_{i0} := 0 }$

For our purpose the values ${\displaystyle a_{ij}}$ then are given by:

${\displaystyle T_{i_{\geq 0} j_{\geq 1}} = \left( \begin{array}{llllll} ~ & 2 & 3 & 4 & \ldots & 10 \\ 11 & 12 & 13 & 14 & \ldots & 20 \\ 21 & 22 & 23 & 24 & \ldots & 30 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ n1 & n2 & n3 & n4 & \ldots & (n + 1)0 \\ \end{array} \right) }$

With that definition the decimal canonical combinatorial number matrix ${\displaystyle T_{ij}}$ has some amazing abilities. The most important one is, that the prime numbers can only appear at certain lattice points, which are given by the columns ${\displaystyle j = 1}$, ${\displaystyle 3}$, ${\displaystyle 7}$ and ${\displaystyle 9}$, while ${\displaystyle i \geq 1}$.

## Properties: decimal canonical combinatorial number matrix

As an overview for ${\displaystyle j = 2}$, ${\displaystyle 4}$, ${\displaystyle 5}$, ${\displaystyle 6}$, ${\displaystyle 8}$, ${\displaystyle 10}$ we have:

${\displaystyle \begin{array}{|l|c|c|c|c||c|c|} \hline & j = 2 & j = 4 & j = 5 & j = 6 & j = 8 & j = 10 \\ \hline & & & & & &\\ \text{Sequence} & 2 \cdot 2 \cdot 3, & 2 \cdot 7, & 3 \cdot 5, & 2 \cdot 2^3 & 2 \cdot 3^2 & 2 \cdot 2 \cdot 5\\ & 2 \cdot 11, & 2 \cdot 2^2 \cdot 3, & 5^2, & 2 \cdot 13 & 2 \cdot 2 \cdot 7 & 2 \cdot 3 \cdot 5 \\ & 2 \cdot 2^4, & 2 \cdot 17, & 5 \cdot 7, & 6^2 & 2 \cdot 19 & 2 \cdot 2^2 \cdot 5 \\ & 2 \cdot 21, & 2 \cdot 2 \cdot 11, & 5 \cdot 3^2 & 2 \cdot 23 & 2 \cdot 2^3 \cdot 3 & 2 \cdot 5^2 \\ & \dots & \dots & \dots & \dots & \dots & \dots \\ \hline & & & & & & \\ \text{Construction law} & n \geq 0: & n \geq 0: & n \geq 2: & n \geq 0: & n \geq 0: & n \geq 1:\\ & & & & & & \\ & (2 \cdot 3 + 5n) \cdot 2 \notin \mathbb{P} & (2 + 5n) \cdot 2 \notin \mathbb{P} & (2n - 1) \cdot 5 \notin \mathbb{P} & (3 + 5n) \cdot 2 \notin \mathbb{P} & (2^2 + 5n) \cdot 2 \notin \mathbb{P} & (5n) \cdot 2 \notin \mathbb{P}\\ \hline \end{array} }$

The proof that all elements of the columns are not prime is trivial, since the two is appearing in each product, which makes it obvious.