An Approach to the Structural Integrity of Prime Number Systems

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

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Section AT-2 - The { Frequency-of-Primes } for [ Modulo M ]

Summary for the Section :

• In this Section , we shall set up another version of the prime-counting function , i.e :
  • the function { PSI-sub-beta-of-(x) } for the [ Modulo M Prime Number System ] ,

    where :

    • the value of [ M ] is the { product-of-primes for the first-[ Q ] prime-numbers } .

  And the function { PSI-sub-beta-of-(x) } shall take on this format :

  • 

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Setting up the set of { Prime Numbers } :

• Let us assume for the time-being that :
  • we were able to identify all prime-numbers up to [ infinity ] ,

  so that :

  • we may now set-up the set { Set [ P ] } ,
    • being the set of prime-numbers arranged in an ascending order .

  We then have :

  • 

    or :

  • 

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The { Product-of-Primes } for the first [ Q ] prime-numbers :

• Let us now set up the quantity [ Q ] , a [ positive integer ] ,
  • being the [ controlling variable ] in the setting up of the { Product-of-Primes } .

  We then set up the the function { rho-of-(Q) } ,

  • with the value of [ rho ] here being the { product-of-primes for the first-[ Q ] prime-numbers } .

  And we have :

  • 

• Let us also set up , at this point , a related function , the function { phi-of-(Q) } :
  • defined by :
    • 

    for our use in the Sections to follow .

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Using { Prodcut-of-Primes } for [ Modulo M ] :

• Let us now set up the [ Modulo M Prime Number System ] ,
  • with the value of [ M ] here being the { product-of-primes } given by :
    • 

  And we take note here again of this characteristic for the [ Modulo M Prime Number System ] ,

  • namely that :
    • 

• As such :
  • the first [ Q ] prime-numbers are always not included as { candidates-for-primes }
    • in the { Modulo M Prime Number System } .

  In other word ,

  • these { first [ Q ] prime-numbers } are omitted from :
    • the set of [ Identifiable Prime Numbers ] ,

      and also :

    • the set of [ Identified Prime Numbers ] ;

    within the context of the [ Modulo M Prime Number System ] .

• Let us now set up the set { Set OMEGA } with [ Q ] elements :
  • 

    to identify these prime-numbers that have been excluded .

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The Adjusted Frequency-of-Primes Function { PSI-sub-beta-of-(x) } :

• Let us now set up the newly adjusted Frequency-of-Primes Function , the function { PSI-of-beta-of-(x) } :
  • 

    for our [ Modulo M Prime Number System ] based on [ Q ] .

  And we note here that :

  • in the nominator of the fraction here :
    • the [ Q ] prime-numbers in { Set OMEGA } have been omitted from the count via the subtraction of [ Q ] ,

    • but the value [ 1 ] is retained to account for the Super-Prime [ 1 ] .

  As such , counting consistency is maintained throughout within the context of the [ Modulo M Prime Number System ] .

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Breaking-up the { Frequency-of-Primes } into 2 components :

• Let us now attempt to express the function { PSI-sub-beta-of-(x) } :
  • 

    as the product of 2 ratios .

  We then set up the ratios [ R1 ] and [ R2 ] as follows :

  • 

    and

  • 

  Consequently , we can now write :

  • 

• Two (2) special things to take note of here :
  • The ratio [ R1 ] is always less than [ 1 ] .

    This is because :

    • the number of prime-numbers identified via the [ Modulo M Prime Number System ]

      can never exceed the number of { candidates-for-prime } for the same [ Modulo M Prime Number System ] .

  • The ratio [ R2 ] is also always less than [ 1 ] .

    This is because :

    • the number of { candidates-for-primes } for the [ Modulo M Prime Number System ]

      is always less the full field of natural numbers from [ 1 ] to [ x ] .

• And we shall be taking a very close look the behavior of the ratio [ R2 ] later in Section AT-4 .

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go to the next Section : Section AT-3 - Deriving the Formula for the { Candidates-for-Primes }

go to the last Section : Section AT-1 - The { Frequency of Prime Numbers } in general

Return to the : Structural Integrity of Prime Number Systems HomePage

Original dated 2012-10-23