On the Unique-Factorization Equation within Prime Number Systems

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

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Section VI - Knocking-Off { Non-Prime Numbers }

Summary for the Section :

• In this Section , we shall set out the general procedure for :
  • knocking-off [ non-prime numbers ] on the list of { candidates-for-primes }
    • based on a set of { Assumed Prime-Numbers } .

  We shall also list-out the procedure for arriving at { Output String of 412-elements } ,

  • based on the same set of { Assumed Prime-Numbers } .

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Determining the Highest { Category of Composite Numbers } that needs to be looked at :

• Let us recall that :
  • [ 7 ] is the smallest prime-number under consideration in the [ Modulo 30 Prime Number System ] .

  As such ,

  • the smallest { Category II Composite Number } that can be formulated with prime-numbers [ 7 ] or above is then :
    • [ 49 ] = 7 x 7 .

  • the smallest { Category III Composite Number } that can be formulated with prime-numbers [ 7 ] or above is then :
    • [ 343 ] = 7 x 7 x 7 .

  • the smallest { Category IV Composite Number } that can be formulated with prime-numbers [ 7 ] or above is then :
    • [ 2401 ] = 7 x 7 x 7 x 7 .

• And since [ 2401 ] is already greater than [ N = 1543 ] , we can now conclude that :
  • { Category IV or above Composite Numbers } will not be involved in the process of

    knocking-off { candidates-for-primes for the [ Modulo 30 prime Number System ] up to [ N = 1543 ] }

    • as [ non-prime numbers ] .

  Consequently , it is only neccessary for us to look at :

  • { Category II Composite Numbers } , and

  • { Category III Composite Numbers } ;

    in the { knocking-off non-prime numbers } process .

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Taking Note of the Special Catch :

• But there is a special catch here , namely that :
  • We are formulating the quote-unquote { composite numbers } here with { assumed prime-numbers } ,
    • and these [ assumed prime-numbers ] may turn-out to be [ prime ] or [ non-prime ] .

  As such , { unique factorization } may-and-may-not work properly ,

  • depending on the circumstances .

• Thus , it is now necessary for us to change tactics .

  And our new procedure here is then :

  • to look at the { 2-elements combinations } that can be formulated with the specific set of { Assumed Prime-Numbers } ,

    instead of :

    • the quote-unquote { Category II Composite Numbers } that can be formulated

      with the said set of { Assumed Prime-Numbers } ;

  • to look at the { 3-elements combinations } that can be formulated with the specific set of { Assumed Prime-Numbers } ,

    instead of :

    • the quote-unquote { Category III Composite Numbers } that can be formulated

      with the said set of { Assumed Prime-Numbers } ;

  • and so-on-and-so-forth as may be necessary .

• And we take note here that :
  • the formula for the Full Count of { 2-elements / 3-elements combinations } can be developed

    via following along the same line-of-logics as presented in the 4th paper of the current series :

    • On Unique-Factorization Combinatorics via the Pascal Triangle .

  And we simply list here the above-said formula without further proof :

  • for the { 2-elements Combinations } :
    • 

      where :

      • [ H ] is the number of { Assumed Prime-Numbers } , and
      • the co-efficients { 1 / 1 } is the 2nd-Row of the Pascal Triangle .

  • for the { 3-elements Combinations } :
    • 

      where :

      • [ H ] is the number of { Assumed Prime-Numbers } , and
      • the co-efficients { 1 / 2 / 1 } is the 3rd-Row of the Pascal Triangle .

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Arriving at the { Output String of 412-elements } :

• To arrive at the { Output String of 412-elements } , we simply :
  • bring-in a duplicate copy of the { Input String of 412-elements } ,

    and

  • evaluate the full sets of { 2-elements combinations } and { 3-elements combinations }
    • arising from the specific set of { Assumed Prime-Numbers }

      and :

      whenever the evaluated value arising therefrom is less than [ N = 1543 ] ,

      • we shall place a [ green-color ball ] into the bin marked with that particular [ evaluated numerical value ] .

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Doing an example next :

• We shall now do an example , next , for clarification purposes .

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go to the next section : Section VII - An Example on Knocking-Off [ Non-Prime Numbers ]

go to the last section : Section V - Setting up the various { Input Strings of 412-elements }

return to the Unique-Factorization Equation HomePage

Original dated 2012-10-08