On the Unique-Factorization Equation within Prime Number Systems

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

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Section V - Setting up the various { Input Strings of 412-elements }

Summary for the Section :

• In this Section , we shall set up the various { Input Strings of 412-elements } .

  And :

  • the total number of variations here is { 2-raised-to-the-[ 409th power ] } .

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Recalling the [ Start-Off Position ] :

• Let us bring-in again the [ Start-Off Position ] from the the last Section :

  

  • noting here again that :
    • [ Bin #1 ] will always spot a [ cyan-color ball ] ,
    • [ Bin #7 ] will always spot a [ red-color ball ] , and
    • [ Bin # 1543 ] will always spot a [ dark-blue-color ball ] .

  And we see here that there are 409 empty bins from [ Bin #11 ] thru to [ Bin #1541 ] ,

  • as per this diagram below :

  

• And for the purpose of combinatorics analysis , we shall now assume that :
  • each of the 409 { candidates-for-primes } associated with these 409 [ empty bins ] is :
    • either :
      • a [ prime-number ] ,

      or :

      • a [ non-prime number ] .

  As such , there are therefore { 2-raised-to-the-[ 409th power ] } different variations/combinations available for us to look at here .

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Creating the [ Input String of 412-elements ] :

• Once we have got a fix on the set of [ Assumed Prime-Numbers ] we wish to investigate for the time-being ,
  • we can then proceed to :
    • put a [ red-color ball ] into each-and-every bin associated with an [ assumed prime-number ]

      in the set of [ Assumed Prime-Numbers ] under our investigation for the time-being .

  The resultant [ 412-bins pattern ] shall then consist of :

  • [ Bin #1 ] spotting a [ cyan-color ball ] ,

  • [ Bin #7 ] spotting a [ red-color ball ] ,

  • [ Bin #11 ] thru [ Bin #1541 ] either spotting a [ red-color ball ] or is [ empty ] ,

    and

  • [ Bin #1543 ] spotting a [ dark-blue-color ball ] .

  And this resultant [ 412-bins pattern ] shall be known as the [ Input String of 412-elements ] .

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Three (3) quick examples of an { Input String of 412-elements } :

• Let us take a look at 3 sample [ Input Strings of 412-elements ] ,
  • for a quick illustration of the mechanics here .

  EXAMPLE I :

  • None of the 409 { candidates-for-primes } are assumed to be [ prime-numbers ] .

  

  EXAMPLE II :

  • All of the 409 { candidates-for-primes } are assumed to be [ prime-numbers ] .

  

  EXAMPLE III :

  • Some of the 409 { candidates-for-primes } are assumed to the [ prime-numbers ] and
    • the actual-and-genuine prime-numbers are correctly identified .

  

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What's next :

• Now that we have a set of [ assumed prime-numbers ] ,
  • we can now proceed to knock-off [ non-prime/composite numbers ] by formulating the quote-unquote [ composite numbers ]

    based on the prime-number [ 7 ] and the said set of [ Assumed Prime-Numbers ] .

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go to the next section : Section VI - Knocking-off [ Non-Prime Numbers ]

go to the last section : Section IV - The a-priori Assumptions and the [ Start-Off Position ]

return to the Unique-Factorization Equation HomePage

Original dated 2012-10-08