# Number Theory/Prime canonical combinatorial number matrix

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The prime canonical combinatorial number matrix, usually shortend to just prime number matrix, is a special arrangement of the natural numbers, used to develop new mathematical tools in number theory. It is a submatrix of the decimal canonical combinatorial number matrix with the same purpose.[1]

## Definition

The prime canonical combinatorial number matrix for the four ${\displaystyle (i > 0)}$ columns:

${\displaystyle \begin{array}{|l|c|c|c|c|} \hline & j = 1 & j = 3 & j = 7 & j = 9 \\ \hline & & & & \\ \text{Sequence} & 11, & 13, & 17, & 19, \\ & 3 \cdot 7, & 23, & 3 \cdot 3^2, & 29, \\ & 31, & 3 \cdot 11, & 37, & 3 \cdot 13, \\ & 41, & 43, & 47, & 7^2, \\ & \dots & \dots & \dots & \dots \\ \hline & & & & \\ \text{Construction law} & n \geq 1: & n \geq 0: & n \geq 0: & n \geq 0: \\ & & & & \\ & 1 + 2 \cdot n \cdot 5 & 3 + 2 \cdot n \cdot 5 & 7 + 2 \cdot n \cdot 5 & 9 + 2 \cdot n \cdot 5 \\ \hline \end{array}}$

is given by:

${\displaystyle P_{\sigma t} = \left( \begin{array}{llll} & ~3 & ~7 & ~9 \\ 11 & 13 & 17 & 19 \\ 21 & 23 & 27 & 29 \\ 31 & 33 & 37 & 39 \\ 41 & 43 & 47 & 49 \\ \vdots & \vdots & \vdots & \vdots \\ n1 & n3 & n7 & n9 \\ \end{array} \right)}$

whereby we quietly assume, that ${\displaystyle 2}$ and ${\displaystyle 5}$ are also part of its cardinality ${\displaystyle \sharp \vert P_{\sigma t} \vert}$.

## Remark

Not all entries in the prime canonical combinatorial number matrix are prime numbers and the total amount of:

${\displaystyle \sharp \vert P_{\sigma t} \vert = \sharp \vert p_n \vert + \mathcal{O}(n_p) \thickapprox 40\%~\text{of}~\sharp \vert T_{ij} \vert }$

primes, plus those who are only odd numbers, are almost forty percent of all numbers in a given interval from one to ${\displaystyle n}$ ${\displaystyle (= \sharp \vert T_{ij} \vert)}$. Thus, if we can get:

${\displaystyle \lim \mathcal{O}(n_p) \to 0 }$

the error term tend to zero, then we are only left with ${\displaystyle \sharp \vert p_n \vert}$, the total amount of prime numbers.

## Properties of prime number matrix

To reach ${\displaystyle \lim \mathcal{O}(n_p) \to 0}$ all (possible) combinations of entry products:

${\displaystyle \operatorname{com}_{\zeta \tau} \operatorname{com}_{\sigma t}~p_{\zeta \tau} p_{\sigma t} \notin \mathbb{P}}$

starting with the first ${\displaystyle -}$, have to be excluded from the canonical combinatorial prime number matrix, whereby the equation is a compactification of:

${\displaystyle \begin{array}{lllll} p_{\zeta \tau} \cdot p_{\sigma t+1} = n_i & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 1} = n_{i + 1} & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 1} = n_{i + 2} & \cdots & n_i + d = n_{i + 1}\\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+2} = n_j & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 2} = n_{j + 1} & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 2} = n_{j + 2} & \cdots & n_j + d = n_{j + 1}\\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+3} = n_k & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 3} = n_{k + 1} & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 3} = n_{k + 2} & \cdots & n_k + d = n_{k + 1}\\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+4} = n_m & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 4} = n_{m + 1} & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 4} = n_{m + 2} & \cdots & n_m + d = n_{m + 1}\\ \end{array}}$

Since it is clear that such a process reveals one underlying pattern in one column of the prime number matrix, it allows therefore to excludes entries from it that are not prime.

### Example

In order to exclude numbers that are not prime, we start with the first entry product and find for each column:

${\displaystyle \begin{array}{lll} j = 1 & n \geq 0: 3 \cdot 7 + 30 \cdot n & \notin \mathbb{P} \\ & & \\ j = 3 & n \geq 1: 3 + 30 \cdot n & \notin \mathbb{P} \\ & & \\ j = 7 & n \geq 0: 3 \cdot 9 + 30 \cdot n & \notin \mathbb{P} \\ & & \\ j = 9 & n \geq 0: 3 \cdot 3 + 30 \cdot n & \notin \mathbb{P} \\ \end{array}}$

eliminating in ${\displaystyle j}$ every third element with a different starting point ${\displaystyle i}$. This is nothing more than:

${\displaystyle \begin{array}{llll} 3 \cdot 3 = 9 & 3 \cdot 13 = 39 & 3 \cdot 23 = 69 & \cdots \\ & & & \\ 3 \cdot 7 = 21 & 3 \cdot 17 = 51 & 3 \cdot 27 = 81 & \cdots \\ & & & \\ 3 \cdot 9 = 27 & 3 \cdot 19 = 57 & 3 \cdot 29 = 87 & \cdots \\ & & & \\ 3 \cdot 11 = 33 & 3 \cdot 21 = 63 & 3 \cdot 31 = 93 & \cdots \\ \end{array}}$

(Please note the constant distance between two multiplications in each line). Those patterns are constructed by the combinations: ${\displaystyle \operatorname{com}_{\sigma t} p_{12} \cdot p_{\sigma t}}$.

### Example

To exclude more numbers that are not prime, this process can be continued with:

${\displaystyle \begin{array}{llll} 7 \cdot 11 = 77 & 7 \cdot 21 = 147 & 7 \cdot 31 = 217 & \cdots \\ & & & \\ 7 \cdot 13 = 91 & 7 \cdot 23 = 161 & 7 \cdot 33 = 231 & \cdots \\ & & & \\ 7 \cdot 17 = 119 & 7 \cdot 27 = 189 & 7 \cdot 37 = 259 & \cdots \\ & & & \\ 7 \cdot 19 = 133 & 7 \cdot 29 = 203 & 7 \cdot 39 = 273 & \cdots \\ \end{array}}$

which points to:

${\displaystyle \begin{array}{lll} j = 1 & n \geq 0: 7 \cdot 13 + 70 \cdot n & \notin \mathbb{P} \\ & & \\ j = 3 & n \geq 1: 7 \cdot 9 + 70 \cdot n & \notin \mathbb{P} \\ & & \\ j = 7 & n \geq 0: 7 \cdot 11 + 70 \cdot n & \notin \mathbb{P} \\ & & \\ j = 9 & n \geq 0: 7 \cdot 7 + 70 \cdot n & \notin \mathbb{P} \\ \end{array}}$

eliminating in ${\displaystyle j}$ every seventh element with a different starting point ${\displaystyle i}$; but only those who are not already divisible by three. Those patterns are constructed by the combinations: ${\displaystyle \operatorname{com}_{\sigma t} p_{13} \cdot p_{\sigma t}}$.

## Remark

The pattern:

${\displaystyle \begin{array}{lllll} 17 \cdot 19 = 323 & 17 \cdot 29 = 493 & 17 \cdot 39 = 663 & \cdots & n_i + 170 = n_{i + 1}\\ & & & & \\ 17 \cdot 21 = 357 & 17 \cdot 31 = 527 & 17 \cdot 41 = 697 & \cdots & n_i + 170 = n_{i + 1}\\ & & & & \\ 17 \cdot 23 = 391 & 17 \cdot 33 = 561 & 17 \cdot 43 = 731 & \cdots & n_i + 170 = n_{i + 1} \\ & & & & \\ 17 \cdot 27 = 459 & 17 \cdot 37 = 629 & 17 \cdot 47 = 799 & \cdots & n_i + 170 = n_{i + 1}\\ \end{array}}$

and ${\displaystyle -}$:

${\displaystyle \begin{array}{lllll} 17^1 = 17 & 17^5 = \dots7 & 17^9 = \dots7 & \cdots & 17^{1+4n} \\ & & & & \\ 17^2 = 289 & 17^6 = \dots9 & 17^{10} = \dots9 & \cdots & 17^{2+4n} \\ & & & & \\ 17^3 = 4913 & 17^7 = \dots3 & 17^{11} = \dots3 & \cdots & 17^{3+4n}\\ & & & & \\ 17^4 = 83.521 & 17^8 = \dots1 & 17^{12} = \dots1 & \cdots & 17^{4+4n} \\ \end{array}}$

are two different things. While all numbers of the second kind ${\displaystyle 1 < k}$, ${\displaystyle n \in \mathbb{N}: n^{k}}$ cannot be prime by defintion ${\displaystyle (\Rightarrow~1 < k \in \mathbb{N}: 17^{k} \notin \mathbb{P})}$ (this is true for the first kind as well), the first pattern establishes in very few steps a residue class that eliminates a lot of entries from ${\displaystyle P_{\sigma t}}$.

## Remark

We point out, that it is recommended to start with ${\displaystyle p_{\sigma t}p_{\sigma t} = p_{\sigma t}^2}$ and:

${\displaystyle \begin{array}{lllll} p_{\zeta \tau} \cdot p_{\sigma t+1} = n_i & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 1} = n_{i + 1} = n_i + \delta & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 1} = n_{i + 2} = n_{i + 1} + \delta & \cdots & n_{i + k} + \delta = n_{i + k + 1}\\ & & & & \\ \hspace{25pt} \Downarrow d_2 & \hspace{56pt} \Downarrow d_2 & \hspace{56pt} \Downarrow d_2 & & \hspace{35pt} \Downarrow d_2 \\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+2} = n_j & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 2} = n_{j + 1} = n_j + \delta & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 2} = n_{j + 2} = n_{j + 1} + \delta & \cdots & n_{j + k} + \delta = n_{j + k + 1}\\ & & & & \\ \hspace{25pt} \Downarrow d_3 & \hspace{56pt} \Downarrow d_3 & \hspace{56pt} \Downarrow d_3 & & \hspace{35pt} \Downarrow d_3 \\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+3} = n_q & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 3} = n_{q + 1} = n_q + \delta & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 3} = n_{q + 2} = n_{q + 1} + \delta & \cdots & n_{q + k} + \delta = n_{q + k + 1}\\ & & & & \\ \hspace{25pt} \Downarrow d_4 & \hspace{56pt} \Downarrow d_4 & \hspace{56pt} \Downarrow d_4 & & \hspace{35pt} \Downarrow d_4 \\ & & & & \\ p_{\zeta \tau} \cdot p_{\sigma t+4} = n_m & p_{\zeta \tau} \cdot p_{\sigma + 1 t + 4} = n_{m + 1} = n_m + \delta & p_{\zeta \tau} \cdot p_{\sigma + 2 t + 4} = n_{m + 2} = n_{m + 1} + \delta & \cdots & n_{m + k} + \delta = n_{m + k + 1}\\ \end{array}}$

that there is not only a specific but constant distant from one element of the row ${\displaystyle n_i}$ to the next ${\displaystyle n_{i + 1}}$ and that there is also a specific constant distance in between the elements of one row ${\displaystyle n_i}$ to the next ${\displaystyle n_j}$.