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 XII - Doing the [ Cut-Off Process ] via the SWING

Summary for the Section :

• In this Section , we shall take a closer look at the [ Cut-Off Process ] ,
  • first appearing in Section X of this paper ,

    where :

    • we were able to formulate 28203 { Category II Composite Numbers } with :
      • an initial set of [ 237 prime-numbers ] running from [ 41 ] thru [ 1531 ] ,

      but only 83 of the 28203 { Category II Composite Numbers } are less than [ N = 1543 ] .

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A Statement of the Problem at-hand :

• Lets us recall that :
  • there are 239 prime-numbers running from [ 7 ] to [ 1531 ] in the [ Modulo 30 Prime-Number-System ] ,

  And for { Category III } , the Split-Line is at [ chi = 0.3333 ] and the split is :

  • [ 2 prime-numbers ] vs. [ 237 prime-numbers ] .

  

  And our [ problem at-hand ] here is to determine :

  • how many of the { Category II Composite Numbers } that can be formulated by the [ 237 prime-numbers ]
    • will come-in at below Upper-Limit of [ N = 1543 ] .

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Identifying the 7 prime-numbers in the region from [ chi = 0.3333 ] to [ chi = 0.5 ] :

• Since we shall be looking at { Category II Category Numbers } arising from the [ 237 prime-numbers ] ,
  • let us now bring-in again the Divsion-Lines for { Category II } and for { Category III } ,
    • for a quick comparison .

  

  

• Let us consolidate the 2 diagrams above ,
  • looking specifically at the region in-between [ chi = 0.3333 ] and [ chi = 0.5000 ] .

  

  And we see here that there are 7 prime-numbers in this region :

  • [ 13 / 17 / 19 / 23 / 29 / 31 / 37 ] .

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Posing the 1st Question :

• The 1st Question we ask here is this :
  • How many { Category II Composite Numbers } can we formulate with the 7 prime-numbers :
    • [ 13 / 17 / 19 / 23 / 29 / 31 / 37 ] ,

      but with each of the formulated-numbers staying below the Upper-Limit of [ N = 1543 ] .

• Since the square of [ 37 ] is less than [ N = 1543 ] ,
  • the answer here can be calculated via straight-forward Unique-Factorization Combinatorics .

  And the answer is then :

  • 

    yielding :

  • 

• The answer to the [ 1st Question ] is therefore [ 28 ] .

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Posing the 2nd Question :

• The 2nd Question we ask here is this :
  • How many { Category II Composite Numbers } can we formulate with
    • one prime-factor coming from the [ Low Group of 7 prime-numbers ] running [ 13 ] thru [ 37 ] ,

      and

    • the other prime-factor coming from the [ High Group of 230 prime-numbers ] running from [ 41 ] thru [ 1531 ] .

    but with each of the formulated-numbers staying below the Upper-Limit of [ N = 1543 ] .

• Let us now do a { SWING Procedure } to arrive at the answer .

  FIRST : we 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 [ 7 cuts ] in the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ] ,

  • being the swung-around [ 7 cuts ] .

  SECOND : Let us now sub-divide the zone in-between [ chi = 0.5 ] and [ chi = 1.0 ]

  • into 8 sub-zones via the swung-around [ 7-cuts ] ,
    • as per this diagram below .

  

  And we shall now label the [ 8 zones ] here as [ Zone 0 ] thru [ Zone 7 ] ,

  • as shown in the diagram above .

  THIRD : 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	0.6506 to 0.6734	118.69 to 140.27	212	127 thru 1531
    Zone 1	0.6141 to 0.6506	90.76 to 118.69	6	97 / 101 / 103 / 107 / 109 / 113
    Zone 2	0.5989 to 0.6141	81.21 to 90.76	2	83 / 89
    Zone 3	0.5729 to 0.5989	67.09 to 81.21	3	71 / 73 / 79
    Zone 4	0.5413 to 0.5729	53.21 to 67.09	3	59 / 61 / 67
    Zone 5	0.5322 to 0.5413	49.77 to 53.21	1	53
    Zone 6	0.5081 to 0.5322	41.70 to 49.77	2	43 / 47
    Zone 7	0.5000 to 0.5081	39.28 to 41.70	1	41
    *******	******************	******************	**********	*************************
    	TOTAL		230

  And for the calculation of the Partial Count [ z ] ,

  • we shall now assign a [ weight ] to each of the [ 8 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 ] , and
    • a [ weight ] of [ 7 ] for [ Zone 7 ] .

  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	0.6506 to 0.9999	118.69 to 1542.99	212	0	0
    Zone 1	0.6141 to 0.6506	90.76 to 118.69	6	1	6
    Zone 2	0.5989 to 0.6141	81.21 to 90.76	2	2	4
    Zone 3	0.5729 to 0.5989	67.09 to 81.21	3	3	9
    Zone 4	0.5413 to 0.5729	53.21 to 67.09	3	4	12
    Zone 5	0.5322 to 0.5413	49.77 to 53.21	1	5	5
    Zone 6	0.5081 to 0.5322	41.70 to 49.77	2	6	12
    Zone 7	0.5000 to 0.5081	39.28 to 41.70	1	7	7
    *******	******************	******************	**********	**********	**********
    			TOTAL		55

• The answer to the [ 2nd Question ] is therefore [ 55 ] .

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The Final Answer :

• Let us now consolidate the answers to both the [ 1st Question ] and the [ 2nd Question ] ,
  • and we have :
    • [ 28 ] + [ 55 ] = [ 83 ]

• The final Answer is therefore [ 83 ] , as expected .

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go to the next section : Section XIII - Division-Lines and Power Expansions

go to the last section : Section XI - Explaining the [ SWING Process ] and the [ Zoning Process ]

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08