On Prime Number Systems and Core Structural Pattern Propagations

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

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Section XIII - Identifying the Core Structural Pattern for [ 7 / 17 / 37 / 47 / 67 / 97 ]

Summary for the Section :

• In this Section , we shall set up the Core Structural Pattern for the six (6) prime-numbers :
  • [ 7 / 17 / 37 / 47 / 67 / 97 ] .

  And in the next 5 Sections to come ,

  • we shall be looking at certain Cross-Category [ Symmetry and Counter Flow features ]

    arising from these 6 prime-numbers .

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Setting up the 6-elements Core Structural Pattern :

• Let us now set up this table below for the Natural Logarithms of the six (6) prime-numbers :

  Value of [ N ]	7	17	37	47	67	97
  LOG ( N )	1.9459	2.8332	3.6109	3.8501	4.2407	4.5747
  ************	************	************	************	************	************	************

  We then have this schematic presentation below ,

  • showing the Core Structural Pattern on a Logarithmic basis .

  

  And we also take note here that :

  • the Geometric Mean of [ 7 / 17 / 37 / 47 / 67 / 97 ] is roughly [ 33.2238 ] .

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Setting up the Core Structural Frame :

• Let us now also set up the Core Structural Frame , { FRAME 1-A } , as shown below :

  

  • for our comparisons in later Sections .

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What is coming Next :

• We take note at the point that [ 13 4490 9559 ] is a { Category VI Composite Number }
  • arising from the product of the 6 prime-numbers ;

    i.e. :

    • [ 13 4490 9559 ] = 7 x 17 x 37 x 47 x 67 x 97

• And when we split the 6 prime-factors here into 2 groups ,
  • we can always do so via either a [ 1 : 5 ] , a [ 2 : 4 ] , or a [ 3 : 3 ] split .

  And :

  • once we have split the 6 prime-factors into the 2 groups , we can then formulate 2 numbers :

    with :

    • the [ 1st number ] here being the product of the prime-factor/s in the [ 1st group ] ;

      and

    • the [ 2nd number ] here being the prodcut of the prime-factor/s in the [ 2nd group ] .

  And :

  • when we multiply the [ 1st number ] and the [ 2nd number ] together ,
    • we always end up with [ 13 4990 9559 ] as the final product .

  And as such ,

  • the Geometric Mean of the 2 said [ numbers ] is always the { square-root of [ 13 4990 9559 ] } .

• Consequently , when we put the [ 1st number ] and the [ 2nd number ] on the [ Logarimthic X-Axis ] ,
  • the said 2 [ numbers ] will always exhibit [ Symmetry about a Center-Line ] ,
    • with the [ Center-Line ] here being set at the value of { N = square-root-of-[ 13 4490 9559 ] } .

  And this is because :

  • { logarithm of the [ 1st number ] } plus { logarithm of the [ 2nd number ] }
    • is always equal to the { logarithm of [ 13 4490 9559 ] } .

• Let us now take a quick look at the [ Symmetry and Counter-Flow features ]
  • arising from the [ 1 : 5 ] / [ 2 : 4 ] / [ 3 : 3 ] splits in the next 3 Sections to follow .

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go to the next section : Section XIV - [ Category V ] vs. [ Category I ] Patterns arising from [ 7 / 17 / 37 / 47 / 67 / 97 ]

go to the last section : Section XII - The Range for each Category of Composite Numbers

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08