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 be the decimal canonical combinatorial number matrix:



with the following properties:



For our purpose the values then are given by:



With that definition the decimal canonical combinatorial number matrix 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 , , and , while .


Properties: decimal canonical combinatorial number matrix

As an overview for , , , , , we have:



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.