On Prime Number Systems and Core Structural Pattern Propagations

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

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Section XVII - Summarizing on [ Symmetry about a Center-Line ]

Summary for the Section :

• In this Section , we shall be summarizing our findings on :
  • the [ 1 : 5 ] / [ 2 : 4 ] / [ 3 : 3 ] splits for the number [ 13 4490 9559 ] ,
    • from the last 3 Sections .

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Re-cap on { Symmetry about a Center-Line } :

• Let us now bring-in this diagram below ,
  • summarizing our work in the last 3 Sections .

  

  And we now make these observations on the above 5 [ Structure Frames ] :

  • [ FRAME 5-Z ] is the mirror-image of [ FRAME 1-A ] ,

  • [ FRAME 4-Y ] is the mirror-image of [ FRAME 2-B ] ,

  • [ FRAME 3-C ] is in-itself-and-by-itself [ Symmetric about a Center-Line ] ;

    and

  • [ FRAME 2-B ] is not a duplicate or expansion of [ FRAME 1-A ] ,

  • [ FRAME 3-C ] is not a duplicate or expansion of [ FRAME 1-A ] and

    [ FRAME 3-C ] is also not a duplicate or expansion of [ FRAME 2-B ] .

  As such ,

  • [ FRAME 1-A ] , [ FRAME 2-B ] and [ FRAME 3-C ] may be now regarded as :
    • part of the [ core structural frames ] instrumental in the propagation of structural patterns

      arising from the 6 prime-numbers [ 7 / 17 / 37 / 47 / 67 / 97 ] .

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Bringing-in the Pascal Triangle :

• Let us recall that the number of elements in :
  • [ FRAME 1-A ] / [ FRAME 2-B ] / [ FRAME 3-C ] / [ FRAME 4-Y ] / [ FRAME 5-Z ]

    are :

  • [ 6 / 15 / 20 / 15 / 6 ] repectively , as shown in the above diagram .

  And the numbers [ 6 / 15 / 20 / 15 / 6 ] are in fact the inner-5 elements

  • of [ Row 7 ] of the Pascal Triangle ,
    • as marked-of in [ cyan-color ] in the diagram on-the-left below .

  

• And we do bring-in again the Pascal Triangle as per the diagram on-the-right above ,
  • for the interested reader .

  And we now ask this question :

  • Suppose we set up a [ Category VII Compsite Number ] , i.e. the number :
    • [ 1709 2851 3993 ] = 7 x 17 x 37 x 47 x 67 x 97 x 127

    then :

    • when we split the 7 prime-factors into 2 groups via the [ 1 : 6 ] / [ 2 : 5 ] / [ 3 : 4 ] splits ,

      how many elements are there in each of the selected :

      • { Category I / VI } / { Category II / V } / { Catgeory III /IV } Structural Frames ? .

    ( Hint : Try the inner-elements for [ Row 8 ] of the Pascal Triangle . )

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go to the next section : Section XVIII - [ Category VI ] Patterns arising from [ 7 / 17 / 37 / 47/ 67 / 97 ]

go to the last section : Section XVI - [ Category III ] Patterns arising from [ 7 / 17 / 37 / 47/ 67 / 97 ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08