On the Unique-Factorization Equation within Prime Number Systems

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

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Section IV - The a-priori Assumptions and the [ Start-Off Position ]

Summary for the Section :

• In this Section , we shall do the following :
  • state the a-priori assumptions for our combinatorics analysis to follow ;

    and

  • set up the [ Start-Off Position ] ahead of creating the [ Input Strings of 412-elements ] ,
    • in the next Section .

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A Statement of the a-priori Assumptions :

• The a-priori assumptions for our combinatorics analysis to follow are then :
  • [ 1 ] is the super-prime , and :
    • [ 1 ] is always not involved in the formation of a [ composite number ] .

  • [ 7 ] is a prime-number , and :
    • [ 7 ] is the smallest prime-number under our consideration in the [ Modulo 30 Prime Number System ] .

  • [ 1543 ] is a prime-number , and :
    • [ 1543 ] is the prime-number we specifically selected as the [ upper-limit ] of our investigation efforts here .

  ( Please refer to Section III of the 2nd paper in the current series ,

  • i.e. the paper :

    On Counting Natural Numbers up to [ N = 1543 ] ,

    • as to how and why the prime-number [ 1543 ] was selected as the Upper-Limit . )

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The Start-Off Position :

• Let us now bring-in the [ 412 Empty Bins ] from the last Section .

  And based on the above a-priori assunptions , we shall now do the following :

  • put a [ cyan-color ball ] into [ Bin #1 ] ,
    • to indicate that [ 1 ] is the Super-Prime .

  • put a [ red-color ball ] into [ Bin #7 ] ,
    • to indicate that [ 7 ] is a prime-number .

  • put a [ dark-blue-color ball ] into [ Bin #1543 ]
    • to indicate that [ 1543 ] is a prime-number and the [ Upper-Limit ] .

  And this shall be our [ Start-Off Position ] ahead of further munipulations to follow .

• We then have this schematic presentation of the above [ Start-Off Position ] ,
  • as per this diagram below .

  

  And the meaning of the differently colored [ color balls ] is explained immediately below .

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General Rules for the [ Color Balls ] :

• In general , we shall be using only four (4) types of [ color balls ] :
  • [ cyan ] , [ dark-blue ] , [ red ] and [ green ] .

  And each type of [ color ball ] shall carry a different meaning :

  • the one-and-only [ cyan-color ball ] is always reserved for the Super-Prime [ 1 ] ,

  • the one-and-only [ dark-blue-color ball ] is always reserved for the Upper-Limit [ 1543 ] ,

  • [ red-color balls ] are reserved for purpose of indicating the possibilities of [ prime numbers ] ,

  • [ green-color balls ] are reserved for purpose of indicating the possibilities of [ non-prime/composite numbers ] .

• And with this in hand , we are now ready to look at the { Input Strings of 412-elements } next .

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

go to the last section : Section III - Setting up the [ 412 Empty Bins ]

return to the Unique-Factorization Equation HomePage

Original dated 2012-10-08