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 - Explaining the [ SWING Process ] and the [ Zoning Process ]

Summary for the Section :

• In this Section , we shall explain the [ SWING Process ] and the [ Zoning Process ] ,
  • via using a blow-by-blow account :
    • on the counting of { Category II-Type B Composite Numbers }

      in the [ All-Natural Prime-Number-System ] up to [ N = 1543 ] .

  And the [ Full Count number ] that we shall be looking for here is [ 372 ] .

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Setting Up for the { SWING process } :

• Lets us recall that :
  • there are 242 prime-numbers running from [ 2 ] to [ 1531 ] in the [ All-Natural Prime-Number-System ] ,

    and

  • the Split of the [ 242 prime-numbers ] for { Category II } is
    • [ 12 prime-numbers ] vs. [ 230 prime-numbers ] .

    ( Dividing-Line at the square-root of [ N = 1543 ] roughly equal to [ 39.28 ] ) .

  

  And we take special note here of the [ 12 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 [ 12 cuts ] in the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ] ,

  • being the swung-around [ 12 cuts ] .

• Let us now sub-divide the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ]
  • into 13 sub-zones via the swung-around [ 12-cuts ] ,
    • as per this diagram below .

  

• And we shall now label the [ 13 zones ] here as [ Zone 0 ] thru [ Zone 12 ] ,
  • as shown in the diagram above .

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Explaining the SWING and the zones :

• Let us recall that :
  • in our original counting process for { Catgeroy II Composite Numbers } in the 2nd paper :
    • On Counting Natural Numbers up to [ N = 1543 ]

    one of the first things we did was :

    • to divide the prime-numbers up to [ N = 1543 ] into 2 sub-sections ,
      • one [ High ] and one [ Low ] ,

      for :

      • each of the prime-numbers [ 2 / 3 / 5 / 7 / 11 / 13 / 17 / 19 / 23 / 29 / 31 / 37 ] ;

      so that :

      • when any of the { prime-numbers in the [ High Section ] } is multiplied by :
        • the specific [ prime-number giving rise to the sub-division ] ,

        the product will always exceed [ N = 1543 ] .

  Specifically , the Division-Line here is given by the value [ Q ] :

  • [ Q ] = [ 1543 ] divided by [ P ]

    where [ P ] is any one of the prime-numbers [ 2 / 3 / 5 / 7 / 11 / 13 / 17 / 19 / 23 / 29 / 31 / 37 ] .

• We then have this table below ,
  • showing the values of [ P ] and the corresponding values of [ Q ] .

    I.D. of CUT- LINE	* *	Before SWING		* *	After SWING
    		Value of Prime Number	chi-of-[ P ]		Value of [ 1543 ] over [ P ]	chi-of-[ Q ]
    ********	*	**********	**********	*	**********	**********
    		P	P-BAR		Q	Q-BAR
    ********	*	**********	**********	*	**********	**********
    Cut #1		2	0.0944		771.50	0.9056
    Cut #2		3	0.1496		514.33	0.8504
    Cut #3		5	0.2192		308.60	0.7808
    Cut #4		7	0.2651		220.43	0.7349
    Cut #5		11	0.3266		140.27	0.6734
    Cut #6		13	0.3494		118.69	0.6506
    Cut #7		17	0.3859		90.76	0.6141
    Cut #8		19	0.4011		81.21	0.5989
    Cut #9		23	0.4271		67.09	0.5729
    Cut #10		29	0.4587		53.21	0.5413
    Cut #11		31	0.4678		49.77	0.5322
    Cut #12		37	0.4919		41.70	0.5081

  And we take special note here that :

  • while the values of [ P ] and [ Q ] do not exhibit symmetry ,
    • the logarithmic values { chi-of-[ P ] } and { chi-of-[ Q ] } do exhibit { Symmetry about a Center-Line } ,
      • namely the { center-line at [ chi = 0.5 ] } .
• As such :
  • when we do the { SWING Process } via the { Swing-Axle at [ chi = 0.5 ] } ,
    • we have effectively converted :
      • the { 12-Cuts arising from the [ P ]'s }

        to

      • the { 12-Cuts arising from the [ Q ]'s }

  Isn't that wonderful .

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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	0.9056 to 0.9999	771.5 to 1542.99	106	773 thru 1531
    Zone 1	0.8504 to 0.9056	514.33 to 771.509	39	521 thru 769
    Zone 2	0.7808 to 0.8504	308.60 to 514.33	34	311 thru 509
    Zone 3	0.7349 to 0.7808	220.43 to 308.60	16	223 thru 307
    Zone 4	0.6734 to 0.7349	140.27 to 220.43	13	149 thru 211
    Zone 5	0.6506 to 0.6734	118.69 to 140.27	4	127 / 131 / 137 / 139
    Zone 6	0.6141 to 0.6506	90.76 to 118.69	6	97 / 101 / 103 / 107 / 109 / 113
    Zone 7	0.5989 to 0.6141	81.21 to 90.76	2	83 / 89
    Zone 8	0.5729 to 0.5989	67.09 to 81.21	3	71 / 73 / 79
    Zone 9	0.5413 to 0.5729	53.21 to 67.09	3	59 / 61 / 67
    Zone 10	0.5322 to 0.5413	49.77 to 53.21	1	53
    Zone 11	0.5081 to 0.5322	41.70 to 49.77	2	43 / 47
    Zone 12	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 [ 13 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 ] ,
    • a [ weight ] of [ 9 ] for [ Zone 9 ] ,
    • a [ weight ] of [ 10 ] for [ Zone 10 ] ,
    • a [ weight ] of [ 11 ] for [ Zone 11 ] , and
    • a [ weight ] of [ 12 ] for [ Zone 12 ] .

  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	0.9056 to 0.9999	771.5 to 1542.99	106	0	0
    Zone 1	0.8504 to 0.9056	514.33 to 771.509	39	1	39
    Zone 2	0.7808 to 0.8504	308.60 to 514.33	34	2	68
    Zone 3	0.7349 to 0.7808	220.43 to 308.60	16	3	48
    Zone 4	0.6734 to 0.7349	140.27 to 220.43	13	4	52
    Zone 5	0.6506 to 0.6734	118.69 to 140.27	4	5	20
    Zone 6	0.6141 to 0.6506	90.76 to 118.69	6	6	36
    Zone 7	0.5989 to 0.6141	81.21 to 90.76	2	7	14
    Zone 8	0.5729 to 0.5989	67.09 to 81.21	3	8	24
    Zone 9	0.5413 to 0.5729	53.21 to 67.09	3	9	27
    Zone 10	0.5322 to 0.5413	49.77 to 53.21	1	10	10
    Zone 11	0.5081 to 0.5322	41.70 to 49.77	2	11	22
    Zone 12	0.5000 to 0.5081	39.28 to 41.70	1	12	12
    *******	******************	******************	**********	**********	**********
    			TOTAL		372

• Yes , a Total Count of [ 372 ] as expected .

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Explaining the Weights for the Zones :

• For the sake of absolute clarity , let us now make these 13 statements in succession :
  • For the prime-numbers in [ Zone 12 ] ,
    • when these prime-numbers are multiplied by any of the 12 prime-numbers :
      • [ 37 / 31 / 29 / 23 / 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 11 ] ,
    • when these prime-numbers are multiplied by any of the 11 prime-numbers :
      • [ 31 / 29 / 23 / 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 10 ] ,
    • when these prime-numbers are multiplied by any of the 10 prime-numbers :
      • [ 29 / 23 / 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 9 ] ,
    • when these prime-numbers are multiplied by any of the 9 prime-numbers :
      • [ 23 / 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 8 ] ,
    • when these prime-numbers are multiplied by any of the 8 prime-numbers :
      • [ 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 7 ] ,
    • when these prime-numbers are multiplied by any of the 7 prime-numbers :
      • [ 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 6 ] ,
    • when these prime-numbers are multiplied by any of the 6 prime-numbers :
      • [ 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 5 ] ,
    • when these prime-numbers are multiplied by any of the 5 prime-numbers :
      • [ 11 / 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 4 ] ,
    • when these prime-numbers are multiplied by any of the 4 prime-numbers :
      • [ 7 / 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 3 ] ,
    • when these prime-numbers are multiplied by any of the 3 prime-numbers :
      • [ 5 / 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 2 ] ,
    • when these prime-numbers are multiplied by any of the 2 prime-numbers :
      • [ 3 / 2 ] ;

      the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 1 ] ,
    • when these prime-numbers are multiplied by the prime-number [ 2 ] :
      • the product will always stay under the Upper-Limit of [ N = 1543 ] .

  • For the prime-numbers in [ Zone 0 ] ,
    • when these prime-numbers are multiplied by any of the 12 prime-numbers :
      • [ 37 / 31 / 29 / 23 / 19 / 17 / 13 / 11 / 7 / 5 / 3 / 2 ] ;

      the product will always exceed the Upper-Limit of [ N = 1543 ] .

• This then explains the reasons for the specific-but-different weights for each of the 13 zones .

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go to the next section : Section XII - Doing the [ Cut-Off Process ] via the SWING

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

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08