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 II - A Quick Review of Unique-Factorization Combinatorics

Summary for the Section :

• In this Section , we shall do a quick review Unique-Factorization Combinatorics
  • involving techniques previously developed in the 3rd paper of the series :
    • On Unique-Factorization Combinatorics via the Pascal Triangle

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The Equation for the Full Count :

• To refresh our memory , let us bring-in again the equations :
  • for Full Count of the [ composite numbers ] that can be formulated
    • with a set of { [ H ] distinct-and-different prime-numbers } .

  And the equations are :

  • For { Category II Composite Numbers } :
    • the Full Count [ z ] is calculated via this equation :
      • 

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

  • For { Category III Composite Numbers } :
    • the Full Count [ z ] is calculated via this equation :
      • 

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

  • For { Category IV Composite Numbers } :
    • the Full Count [ z ] is calculated via this equation :
      • 

        where the coefficients [ 1 / 3 / 3 / 1 ] is the [ 4th row ] of the Pascal Triangle .

  • And so-on-and-so-forth .

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Calculation of the Full Fields :

• In the 2 Sections to follow immediately , we shall be bringing-in the summary data :
  • for :
    • the [ All-Natural Prime Number System ] up to [ N = 1543 ] , and

    • the [ Modulo 30 Prime Number System ] up to [ N = 1543 ] ;

    on a by-category-and-by-type basis .

  It is therefore necessary for us to set up the calculation process :

  • for [ Full Fields ] on a { by-category-and-by-type } basis ,

    so that :

    • we can also take a look at the [ Hit Percentages ] on the same { by-category-and-by-type } basis .

  And the term [ Hit Percentage ] here is simply defined as :

  • the percentage of the Full Field of composite-numbers in each particular [ category-and-type ]
    • that are :
      • less than [ N = 1543 ]

        and

      • therefore successful in knocking-off a { candidate-for-prime } from the original proposed { List of Prime-Numbers } .

• It is now appropriate for us :
  • to do an example to illustrate what we meant by the term [ Full Field ] .

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An example of the [ Full Field ] :

• We shall now do an example for the [ Full Field ] for :
  • { Category V-Type C Composite Numbers } for the [ All-Natural Prime-Number-System ] up to [ N = 1543 ] .

  And here is the Example :

  • First , let us recall that there are 242 prime-numbers ,
    • running from [ 2 ] thru [ 1531 ] ,

      at our disposal for the [ All-Natural Prime Number System ] up to [ N = 1543 ] .

    And for { Category V Composite Numbers } , the Division-Line is set at the [ 5th-root ] of [ 1543 ] ,

    • roughly equal to 4.3418 ,

    so that :

    • the [ 242 prime-numbers ] may now be split into two (2) sub-groups :
      • the [ LOW Group ] consisting of the 2 prime-numbers [ 2 ] and [ 3 ] , and

      • the [ HIGH Gourp ] consisting of 240 prime-numbers running from [ 5 ] thru [ 1531 ] .

  • Secondly , let us recall that { Category V - Type C Composite Numbers } are [ composite numbers ] :
    • with 3 prime-factors in the [ LOW Group ] and 2 prime-factors in the [ HIGH Group ] .

    Let us now calculate the number of { Category III Composite Numbers }

    • that can be formulated with the [ 2 prime-numbers ] in the [ LOW Group ] ;

      and we have :

      • 

        yielding :

      • 

    Let us also calculate the number of { Category II Composite Numbers }

    • that can be formulated with the [ 240 prime-numbers ] in the [ HIGH Group ] ;

      and we have :

      • 

        yielding :

      • 

  • And the Full Field for { Category V-Type C Composite Numbers } for the { All-Natural Prime Number System } up to [ N = 1543 ]
    • is then :
      • 

• Let's move on .

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go to the next section : Section III - Summarizing data on the [ All-Natural Prime Number System ]

go to the last section : Section I - Introduction

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08