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

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

Top Of Page

Section VII - { Category II-Type B } in the [ Modulo 30 Prime Number System ]

Summary for the Section :

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

  And the [ Full Count number ] that we shall be looking for here is [ 112 ] .

go to Top Of Page

Setting Up for the { SWING process } :

• Lets us recall that :
  • the Split of the [ 239 prime-numbers ] for { Category II } is
    • [ 9 prime-numbers ] vs. [ 230 prime-numbers ] .

  

  And we take special note here of the [ 9 cuts ] in the zone in-between [ chi = 0.0 ] and [ chi = 0.5 ]

  • 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 [ 9 cuts ] in the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ] ,

  • being the swung-around [ 9 cuts ] .

• Let us now sub-divide the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ]
  • into 10 sub-zones via the swung-around [ 9-cuts ] ,
    • as per this diagram below .

  

• And we shall now label the [ 10 zones ] here as [ Zone 0 ] thru [ Zone 9 ] ,
  • as shown in the diagram above .

go to Top Of Page

Calculating the Full Count :

• Let recall that :
  • there are originally [ 230 prime-numbers ] in the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ] .

  

  Let us see now how these [ 230 prime-numbers ] would fall into each of the [ 10 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	0.7349 to 0.9999	220.43 to 1542.99	195	223 thru 1531
    Zone 1	0.6734 to 0.7349	140.27 to 220.43	13	149 thru 211
    Zone 2	0.6506 to 0.6734	118.69 to 140.27	4	127 / 131 / 137 / 139
    Zone 3	0.6141 to 0.6506	90.76 to 118.69	6	97 / 101 / 103 / 107 / 109 / 113
    Zone 4	0.5989 to 0.6141	81.21 to 90.76	2	83 / 89
    Zone 5	0.5729 to 0.5989	67.09 to 81.21	3	71 / 73 / 79
    Zone 6	0.5413 to 0.5729	53.21 to 67.09	3	59 / 61 / 67
    Zone 7	0.5322 to 0.5413	49.77 to 53.21	1	53
    Zone 8	0.5081 to 0.5322	41.70 to 49.77	2	43 / 47
    Zone 9	0.5000 to 0.5081	39.28 to 41.70	1	41
    *******	******************	******************	**********	*************************
    	TOTAL		230

• For the calculation of the Full Count [ z ] ,
  • we shall now assign a [ weight ] to each of the [ 10 zones ] :
    • a [ weight ] of [ 0 ] for [ Zone 0 ] ,
    • a [ weight ] of [ 1 ] for [ Zone 1 ] ,
    • a [ weight ] of [ 2 ] for [ Zone 2 ] ,
    • a [ weight ] of [ 3 ] for [ Zone 3 ] ,
    • a [ weight ] of [ 4 ] for [ Zone 4 ] ,
    • a [ weight ] of [ 5 ] for [ Zone 5 ] ,
    • a [ weight ] of [ 6 ] for [ Zone 6 ] ,
    • a [ weight ] of [ 7 ] for [ Zone 7 ] ,
    • a [ weight ] of [ 8 ] for [ Zone 8 ] , and
    • a [ weight ] of [ 9 ] for [ Zone 9 ] .

  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	0.7349 to 0.9999	220.43 to 1542.99	195	0	0
    Zone 1	0.6734 to 0.7349	140.27 to 220.43	13	1	13
    Zone 2	0.6506 to 0.6734	118.69 to 140.27	4	2	8
    Zone 3	0.6141 to 0.6506	90.76 to 118.69	6	3	18
    Zone 4	0.5989 to 0.6141	81.21 to 90.76	2	4	8
    Zone 5	0.5729 to 0.5989	67.09 to 81.21	3	5	15
    Zone 6	0.5413 to 0.5729	53.21 to 67.09	3	6	18
    Zone 7	0.5322 to 0.5413	49.77 to 53.21	1	7	7
    Zone 8	0.5081 to 0.5322	41.70 to 49.77	2	8	16
    Zone 9	0.5000 to 0.5081	39.28 to 41.70	1	9	9
    *******	******************	******************	**********	**********	**********
    			TOTAL		112

• Yes , a Total Count of [ 112 ] as expected .

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

go to Top Of Page

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

go to the last section : Section VI - { Category II-Type A } in the [ Modulo 30 Prime Number System ]

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08