On the Match-Up of Structural Patterns within Prime Number Systems

by Frank C. Fung ( 1st published in October, 2012. )

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Section XI - { Category III-Type C } in the [ Modulo 30 Prime Number System ]

Summary for the Section :

• In this Section , we shall take another look at :
  • { Category III-Type C Composite Numbers } in the [ Modulo 30 Prime Number System ] .

  And the Full Count number that we shall be looking for here is [ 1 ] .

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Setting up the Cut-Off on { Category II Composite Numbers } :

• Lets us recall that :
  • the Split of the [ 239 prime-numbers ] for { Category III } is
    • [ 2 prime-numbers ] vs. [ 237 prime-numbers ] .

  

  And the first thing we need to do here is to set up the { Category II Composite Numbers }

  • arising from the [ 237 prime-numbers ] .

  And we can now utilize the Unique-Factorization Combinatorics equation previously established in the 4th paper of the series :

  • On Unique-Factorization Combinatorics via the Pascal Triangle

    to establish that :

    • a total of 28203 { Category II Composite Numbers } can be formulated with [ 237 prime-numbers ] .

    i.e. :

    • 

      where the coefficients [ 1 / 1 ] is the 2nd Row of the Pascal Triangle .

    and consequently :

    • 

  We then have this diagram below ,

  • showing the range for the 28203 { Category II Composite Numbers }
    • running from [ chi = 0.6667 ] thru [ chi = 2.0000 ] .

  

• And we need to establish now , via this multiplication table below , that :
  • only 83 of the 28203 { Category II Composite Numbers } are less than [ N = 1543 ] .

• And the 83 { Category II Composite Numbers } less than [ N = 1543 ] are marked-off in [ white-color ] in the above table .

  ( We shall provide further detals on the Cut-Off process later in Section XII . )

• We then have this schematic presentation below ,
  • for the range of the said 83 { Category II Composite Numbers } less than [ N = 1 543 ] .

  

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Setting up for the SWING :

• Let us now set up the [ Swing Axle ] at [ chi = 0.5 ] ,
  • as shown in the diagram below .

  

  We can then proceed to do the [ SWING ] using the [ Swing Axle ] ,

  • to arrive at this diagram below :

  

  And we see here that there are now [ 2 cuts ] in the zone in-between [ chi = 0.6667 ] and [ chi = 1.0 ] ,

  • being the swung-around [ 2 cuts ] .

• Let us now sub-divide the zone in-between [ chi = 0.6667 ] and [ chi = 1.0 ]
  • into 3 sub-zones via the swung-around [ 2-cuts ] ,
    • as per this diagram below .

  

• And we shall now label the [ 3 zones ] here as [ Zone 0 ] thru [ Zone 2 ] ,
  • as shown in the diagram above .

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Calculating the Full Count :

• Let recall that :
  • there are originally [ 83 composite-numbers ] in the zone in-between [ chi = 0.6667 ] and [ chi = 1.0 ] .

  

  Let us see now how these [ 83 composite-numbers ] would fall into each of the [ 3 zones ] ,

  • and we have this table below showing the said statistics .

    	Range for the Zone		Number of Composite -Numbers in the Zone	Comments
    	Logarithmic Scale via the CHI-Axis	Real Number Scale
    Zone 0	0.7349 to 0.9999	220.42 to 1542.99	82	221 thru 1541
    Zone 1	0.6734 to 0.7349	140.27 to 220.42	1	169
    Zone 2	0.6667 to 0.6734	133.53 to 140.37	0	none
    *******	******************	******************	**********	**************************
    	TOTAL		83

• For the calculation of the Full Count [ z ] ,
  • we shall now assign a [ weight ] to each of the [ 3 zones ] :
    • a [ weight ] of [ 0 ] for [ Zone 0 ] ,
    • a [ weight ] of [ 1 ] for [ Zone 1 ] , and
    • a [ weight ] of [ 2 ] for [ Zone 2 ] .

  We then have this table below ,

  • showing the actual calculation process .

    	Range for the Zone		Number of Composite -Numbers in the Zone	Weight Factor	Roving Count
    	Logarithmic Scale via the CHI-Axis	Real Number Scale
    	- - -	- - -	Q	W	Q x W
    Zone 0	0.7349 to 0.9999	220.42 to 1542.99	82	0	0
    Zone 1	0.6734 to 0.7349	140.27 to 220.42	1	1	1
    Zone 2	0.6667 to 0.6734	133.53 to 140.72	0	2	0
    *******	******************	******************	**********	**********	**********
    			TOTAL		1

• Yes , a Full Count of [ 1 ] as expected , but lots of work to get here .

  ( Full explanation of the { SWING priocess } and the { Zoning Process } in Section XI . )

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go to the next section : Section XI - Explaining the { SWING Process } and the { Zoning Process }

go to the last section : Section IX - { Category III-B } in the [ Modulo 30 Prime Number System ]

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Original dated 2012-10-08