On Prime Number Systems and Core Structural Pattern Propagations

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

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Section VII - Derived Structural Patterns for [ Category II Composite Numbers ]

Summary for the Section :

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

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Structural Patterns for { Category II Composite Numbers ] :

• Let us quickly state that :
  • there are exactly 10 distinct-and-different { Category II Composite Numbers } here ,

  

  and

  • the Geometric Mean of these 10 Composite-Numbers is [ 454.91 ] .

• Let us now quickly split the said 10 Composite-Numbers into two (2) sub-groups ,
  • [ Set A ] and [ Set B ] respectively .

  For [ Set A ] , we have the squares of [ 7 / 17 / 37 / 47 ] :

  • [ 49 ] = 7 x 7
  • [ 289 ] = 17 x 17
  • [ 1369 ] = 37 x 37
  • [ 2209 ] = 47 x 47

  For [ Set B ] , we have remainder 6 elements :

  • [ 119 ] = 7 x 17
  • [ 259 ] = 7 x 37
  • [ 329 ] = 7 x 47
  • [ 629 ] = 17 x 37
  • [ 799 ] = 17 x 47
  • [ 1739 ] = 37 x 47

• Let us now bring-in the Natural Logarithms for both [ Set A ] and [ Set B ] :
  • For [ Set A ] :

    	SET A
    Value of [ N ]	49	289	1369	2209
    LOG(N)	1.7567	2.4666	3.0887	3.2801
    ************	************	************	************	************

  • For [ Set B ] :

    	SET B
    Value of [ N ]	119	259	329	629	799	1739
    LOG(N)	2.1116	2.4227	2.5184	2.7777	2.8733	3.1844
    ************	************	************	************	************	************	************

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

  

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Setting up the Structural Frames :

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

  

  

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

go to the last section : Section VI - Selecting the Core Structural Pattern for our Demonstrations

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08