On the Match-Up of Structural Patterns within Prime Number Systems

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

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Section VIII - { Category III-Type A } in the [ Modulo 30 Prime Number System ]

Summary for the Section :

• In this Section , we shall take another look at :
  • { Category III-Type A Composite Numbers } in the [ Modulo 30 Prime Number System ] .

  And the [ Full Count number ] that we shall be looking for here is [ 4 ] .

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Setting Up [ Division-Line ] :

• Lets us recall that :
  • there are 239 prime-numbers running from [ 7 ] thru [ 1531 ] ,

    at our disposal in the [ Modulo 30 Prime Number System ] up to [ N = 1543 ] .

  And for { Category III } :

  • the [ Divison-Line ] is set at the { cube-root of [ 1543 ] } ,
    • roughly equal to [ 11.56 ] .

  

  The split of the [ 239 prime-numbers ] here for { Category III } is then :

  • 2 prime-numbers running from [ 7 ] to [ 11 ]

    vs.

  • 237 prime-numbers running from [ 13 ] thru [ 1531 ] .

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Calculating the Full Count :

• Since the cube of [ 11 ] is less than [ N = 1543 ] ,
  • { Category III Composite Numbers } formulated from the { 2 prime-numbers running from [ 7 ] to [ 11 ] }

    are also less than [ N = 1543 ] .

  As such ,

  • the Full Count [ z ] for { Category III-Type A } can be readily calculated via Unique-Factorization Combinatorics .

• We then have :
  • 

    where the coefficients [ 1 / 2 / 1 ] is the [ 3rd row ] of the Pascal Triangle .

  consequently yielding :

  • 

• Again , rather simple and straight-forward .

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go to the next section : Section IX - { Category III-Type B } in the [ Modulo 30 Prime Number System l

go to the last section : Section VII - { Category II-Type B } in the [ Modulo 30 Prime Number System l

return to the Matching-Up Structural Patterns HomePage

Original dated 2012-10-08