On Prime Number Systems and Core Structural Pattern Propagations

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

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Section IX - Derived Structural Patterns for [ Category IV Composite Numbers ]

Summary for the Section :

• In this Section , we shall look at [ Category IV Composite Numbers ] arising from the prime-numbers [ 7 / 17 / 37 / 47 ] .

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Structural Patterns for { Category IV Composite Numbers } :

• Let us quickly state that :
  • there are exactly 35 distinct-and-different { Category IV Composite Numbers } here ,

  

  and

  • the Geometric Mean of these 35 Composite-Numbers is exactly [ 20 6941 ] ,
    • with [ 20 6941 ] being equal to [ 7 x 17 x 37 x 47 ] .

• Let us now quickly split the said 35 Composite-Numbers into four (4) sub-groups ,
  • [ Set A ] , [ Set B ] , [ Set C ] and [ Set D ] respectively .

  For [ Set A ] with 16 elements ,

  • the Natural Logarithms here are then :

    	SET A
    Value of [ N ]	2401	5831	1 2691	1 6121
    LOG(N)	7.7836	8.6709	9.4486	9.6879
    ************	************	************	************	************
    Value of [ N ]	3 4391	8 3521	18 1781	23 0911
    LOG(N)	10.4456	11.3329	12.1106	12.4598
    ************	************	************	************	************
    Value of [ N ]	35 4571	86 1101	187 4161	238 0961
    LOG(N)	12.7787	13.6660	14.4437	14.6829
    ************	************	************	************	************
    Value of [ N ]	72 6761	176 4991	384 1451	487 9681
    LOG(N)	13.4963	14.3837	15.1614	15.4006
    ************	************	************	************	************

  For [ Set B ] with 6 elements ,

  • the Natural Logarithms here are then :

    	SET B
    Value of [ N ]	1 4161	6 7081	10 8241	39 5641	63 8401	302 4121
    LOG(N)	9.5582	11.1137	11.5921	12.8883	13.3667	14.9221
    ************	************	************	************	************	************	************

  We then have this diagram below showing the structural patterns for both [ Set A ] and [ Set B ] :

  

  For [ Set C ] with 12 elements ,

  • the Natural Logarithms here are then :

    	SET C
    Value of [ N ]	3 0821	7 4851	16 2911	20 6941
    LOG(N)	10.3360	11.2233	12.0010	12.2402
    ************	************	************	************	************
    Value of [ N ]	3 9151	9 5081	20 6941	26 2871
    LOG(N)	10.5752	11.4625	12.2402	12.4794
    ************	************	************	************	************
    Value of [ N ]	8 5211	20 6941	45 0401	57 2131
    LOG(N)	11.3529	12.2402	13.0179	13.2571
    ************	************	************	************	************
    Value of [ N ]	20 6941	50 2571	109 3831	138 9461
    LOG(N)	12.2402	13.1275	13.9052	14.1444
    ************	************	************	************	************

  For [ Set D ] with 1 single element :

  • the Natural Logarithm here is then :

    	Set D
    Value of [ N ]	20 6941
    LOG(N)	12.2402
    ************	************

  We then have this diagram below showing the structural patterns for both [ Set C ] and [ Set D ] :

  

  • And we note here that the pre-dominant feature in the above diagram is [ FRAME 1-A ] .

  Let us now look at another presentation of both [ Set C ] and [ Set D ] as follows :

  

  • And the pre-dominant feature here in the above diagram is [ FRAME 3-Z ] ,
    • the mirror-image of [ FRAME 1-A ] .

  Let us now look at yet another presentation of both [ Set C ] and [ Set D ] , as shown below :

  

  • And the pre-dominant feature here is [ symmetry about a Center-Line ] , namely :
    • the line at [ LOG ( 206941 ) ] = 12.2402 .

  Intriguing , isn't it !

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Setting up the Structural Frames :

• Let us also set up , at this point , the Structural Frames [ FRAME 4-A ] and [ FRAME 4-B ] ,
  • for our comparisons later in Section XI ,
    • as per this set of 2 diagrams below .

  

  

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A Special Comment here :

• We take special notice here that :
  • we originally started-out with 4 prime-numbers , namely [ 7 / 17 / 37 / 47 ] .

  In this regard , [ Category IV ] represented a special threshold here ,

  • and the number [ 206941 = 7 x 17 x 37 x 47 ] must necessarily show up here at [ Category IV ] .

• We shall come back to this comment here later in Section XII thru XVIII ,
  • when we do the Structural Patterns again for a [ 6 prime numbers ] set , namely :
    • the set [ 7 / 17 / 37 / 47 / 67 / 97 ] .

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go to the next section : Section X - Derived Structural Patterns for [ Category V Composite Numbers ]

go to the last section : Section VIII - Derived Structural Patterns for [ Category III Composite Numbers ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08