On Prime Number Systems and Core Structural Pattern Propagations

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

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Section XVIII - [ Category VI ] Patterns arising from [ 7 / 17 / 37 / 47 / 67 / 97 ]

Summary for the Section :

• In this Section , we shall take a look at :
  • a special-and-specific [ Category VI Composite Number ] ,

    namely the number :

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

  and

  • the Structural Pattern for 140 selected [ Category VI Composite Numbers ] ,
    • displaying [ symmetry about a Center-Line ] .

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Stating the Basic Statistics for [ Category VI Composite Numbers ] :

• We state here , first-of-all and for the record , that :
  • the total number of [ Category VI Composite Numbers ] arising from 6 prime-numbers is [ 462 ] .

  And as to { How we arrive as this Total } ,

  • this will be full discussed and quantified in the 4th paper in the current series , namely the paper :
    • On Unique-Factorization Combinatorics via the Pascal Triangle )

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Setting up the constants [ K ] and [ K-SQUARE ] :

• Let us now set up the Constant [ K ] :
  • [ K ] = [ 13 4490 9559 ] = 7 x 17 x 37 x 47 x 67 x 97

  And we note here that :

  • [ K ] is the [ Product-of-Primes ] for the 6 prime-numbers [ 7 / 17 / 37 / 47 / 67 / 97 ] .

    and

  • [ K ] is a [ Category VI Composite Number ] .

• We shall now also set up another Constant [ K-SQUARE ] , given by :
  • [ K-SQUARE ] = [ 180 8781 7218 8957 4481 ] = 7 x 7 x 17 x 17 x 37 x 37 x 47 x 47 x 67 x 67 x 97 x 97

    for our use immediately below .

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Setting up the 140 [ Category VI Composite Numbers ] :

• Let us recall that :
  • [ K-SQUARE ] is the product of 12 prime-numbers .

  Let us now break-up the 12 prime-numbers into 2 groups ,

  • with each group containing 6 prime-numbers .

  We then set up :

  • the [ 1st number ] being the product of the 6 prime-numbers in the [ 1st Group ] ,

    and

  • the [ 2nd number ] being the product of the 6 prime-numbers in the [ 2nd Group ] ,

  Interesting enough :

  • when we multiply the [ 1st number ] and the [ 2nd number ] together ,
    • the product here is always [ K-SQUARE ] ;

      and

    • the Geometric Mean of the 2 [ numbers ] is then always [ K = 13 4990 9559 ] .

• This then allows for the set up of :
  • the 70 pairs of { Category VI Composite Numbers } exhibiting [ Symmetry about a Center-Line ] ,
    • when each pair of [ numbers ] is put onto the [ Logarithmic X-Axis ] , same as before .

  Let us now take a quick look at the counting of the 140 { combinations } here , next .

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Taking a first look at the combinatorics of { Catgeory VI Composite Numbers } :

• We then set up this table below , showing the combinatorics counts for :
  • the Full Count of 462 { Category VI Composite Numbers } arising from the [ 6 prime-numbers ] , and

  • the Partial Count of 140 { Category VI Composite Numbers } exhibiting { Symmetry about a Center-Line } .

    Characteristics of the Number		Combinatoris Count
    Number of primes as prime-factors	Nominal Format of the powers of Prime Factors	Combinations for the Full Count	Roving Count for the 70 pairs
    ***************	***************	***************	***************
    1	6	6	x
    2	5-1	30	x
    	4-2	30	x
    	3-3	15	x
    3	4-1-1	60	x
    	3-2-1	120	x
    	2-2-2	20	20
    4	3-1-1-1	60	x
    	2-2-1-1	90	90
    5	2-1-1-1-1	30	30
    6	1-1-1-1-1-1	1	x
    ***************	***************	***************	***************
    Total		462	140

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go to the next section : Section XIX - Concluding Remarks

go to the last section : Section XVII - Summarizing on [ Symmetry about a Center-Line ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08