S6. Details on { Category III Odd-Numbers }

An Approach to Counting Odd Numbers

Section VI - Details on { Category III Odd-Numbers }

In this Section , we shall provide further details on { Category III Odd-Numbers } .

Let us first bring-back the statistics for this category , from Section IV :

Category
of
Odd
Number Number
of
members
in
category *
*
* Smallest
member
in
category Largest
member
in
category Largest member
in category
factorized

Category III 2,451 27 19,677 3 x 7 x 937
Let us start-off by setting up a sufficiently-large { 3-Dimensional Matrix } , say of size [ 1,000 x 1,000 x 1,000 ] :
- so that the value of the [ element ] at { position [ i ] , [ j ] , [ k ] } is given by :
  - where [ OP-i ] , [ OP-j ] and [ OP-k ] are { Odd Prime Numbers } .
We note here that :
- For [ i ] = [ j ] = [ k ] :
  - and this is an { element } on the diagonal of the matrix .
- For any two 2 members of { i , j , k } being the same and the third member being different , we have :
  - so that the { 3-Dimensional Matrix } is [ tri-symmetric ] at certain locations .
- For the three (3) members of { i , j , k } all being different , we have :
  - so that the { 3-Dimensional Matrix } is [ hexa-symmetric ] at certain locations .
And we simply note here that we can go thru the same type of procedure as outlined in the previous Section ,
- to determine , for inclusion into our [ count ] , as to which [ elements ] are less than or equal to [ 19,683 ] .

Category of Odd Number	Number of members in category	* * *	Smallest member in category	Largest member in category	Largest member in category factorized
Category III	2,451		27	19,677	3 x 7 x 937

As we move forward onto { Category IV } and beyond :
- the { combinatorics } will get more complicated , and
  
  higher-ordered [ symmetries ] at [ certain locations ] will also be involved.