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 columns:



is given by:



whereby we quietly assume, that and are also part of its cardinality .


Remark

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



primes, plus those who are only odd numbers, are almost forty percent of all numbers in a given interval from one to . Thus, if we can get:



the error term tend to zero, then we are only left with , the total amount of prime numbers.


Properties of prime number matrix

To reach all (possible) combinations of entry products:



starting with the first , have to be excluded from the canonical combinatorial prime number matrix, whereby the equation is a compactification of:



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:



eliminating in every third element with a different starting point . This is nothing more than:



(Please note the constant distance between two multiplications in each line). Those patterns are constructed by the combinations: .


Example

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



which points to:



eliminating in every seventh element with a different starting point ; but only those who are not already divisible by three. Those patterns are constructed by the combinations: .

Remark

The pattern:



and :



are two different things. While all numbers of the second kind , cannot be prime by defintion (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 .

Remark

We point out, that it is recommended to start with and:



that there is not only a specific but constant distant from one element of the row to the next and that there is also a specific constant distance in between the elements of one row to the next .