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 IX - { Category III-Type B } in the [ Modulo 30 Prime Number System ]

Summary for the Section :

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

  And the [ Full Count number ] that we shall be looking for here is [ 9 ] .

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Setting Up for the SWING :

• 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 2 prime-numbers [ 7 ] and [ 11 ] .

  We then set up :

  • [ 49 ] = 7 x 7
  • [ 77 ] = 7 x 11
  • [ 121 ] = 11 x 11

  

  And we take special note here of the [ 3 cuts ] in the zone in-between [ chi = 0.0 ] and [ chi = 0.6667 ]

  • as marked-off as the [ 49 / 77 / 121 Lines ] in the bottom strip in the diagram above .

• 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 [ 3 cuts ] in the zone in-between [ chi = 0.3333 ] and [ chi = 1.0 ] ,

  • being the swung-around [ 3 cuts ] .

• Let us now sub-divide the zone in-between [ chi = 0.3333 ] and [ chi = 1.0 ]
  • into 4 sub-zones via the swung-around [ 3-cuts ] ,
    • as per this diagram below .

  

• And we shall now label the [ 4 zones ] here as [ Zone 0 ] thru [ Zone 3 ] ,
  • as shown in the diagram above .

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Calculating the Full Count :

• Let recall that :
  • there are originally [ 237 prime-numbers ] in the zone in-between [ chi = 0.3333 ] and [ chi = 1.0 ] .

  

  Let us see now how these [ 237 prime-numbers ] would fall into each of the [ 4 zones ] ,

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

    	Range for the Zone		Number of Primes in the Zone	Comments
    	Logarithmic Scale via the CHI-Axis	Real Number Scale
    Zone 0	0.4699 to 0.9999	31.49 to 1542.99	231	37 thru 1531
    Zone 1	0.4083 to 0.4699	20.04 to 31.49	3	23 / 29 / 31
    Zone 2	0.3468 to 0.4083	12.75 to 20.04	3	13 / 17 / 19
    Zone 3	0.3333 to 0.3468	11.56 to 12.75	0	none
    *******	******************	******************	**********	**************************
    	TOTAL		237

• For the calculation of the Full Count [ z ] ,
  • we shall now assign a [ weight ] to each of the [ 4 zones ] :
    • a [ weight ] of [ 0 ] for [ Zone 0 ] ,
    • a [ weight ] of [ 1 ] for [ Zone 1 ] ,
    • a [ weight ] of [ 2 ] for [ Zone 2 ] , and
    • a [ weight ] of [ 3 ] for [ Zone 3 ] .

  We then have this table below ,

  • showing the actual calculation process .

    	Range for the Zone		Number of Primes in the Zone	Weight Factor	Roving Count
    	Logarithmic Scale via the CHI-Axis	Real Number Scale
    	- - -	- - -	Q	W	Q x W
    Zone 0	0.4699 to 0.9999	31.49 to 1542.99	230	0	0
    Zone 1	0.4083 to 0.4699	20.04 to 31.49	3	1	3
    Zone 2	0.3468 to 0.4083	12.75 to 20.04	3	2	6
    Zone 3	0.3333 to 0.3468	11.56 to 12.75	0	3	0
    *******	******************	******************	**********	**********	**********
    			TOTAL		9

• Yes , a Total Count of [ 9 ] as expected .

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

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go to the next section : Section X - { Category III-C } in the [ Modulo 30 Prime Number System ]

go to the last section : Section VIII - { Category III-A } [ Modulo 30 Prime Number System ]

return to the Matching-Up Structural Patterns HomePage

Original dated 2012-10-08