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-3 - Deriving the Formula for the { Candidates-for-Primes }

Summary for the Section :

• In this Section , we shall attempt to answer this question :
  • For the { Modulo M Prime Number Systems } involving the { Product-of-Primes } as the modulo's :
    • how will the total number of { candidates-for-primes } be affected ,
      • when we move from a lower [ Modulo M Prime Number System ] to the next-higher [ Modulo M Prime Number System ] ?

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A Quick Illustration of the Expansion Mechanism :

• Let us recall that for the [ Modulo 30 Prime Number System ] :
  • [ 30 ] = 2 x 3 x 5 ;

  • the 8 candidates [ 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 ] are not divisible by [ 2 ] , [ 3 ] , or [ 5 ] ;

  • the smallest prime-number under consideration is the next prime-number , [ 7 ] .

  Thus , for the expansion from [ Modulo 30 ] to [ Modulo 210 ] , we can simply :

  • take the 8 candidates-for-primes for [ modulo 30 ] ,

    and

  • add to each of the 8 candidates the values [ 0 / 30 / 60 / 90 / 120 / 150 / 180 ]
    • to arrive at the 56 [ New Possibilities ] for the { Candidates-for-Primes } for [ modulo 210 ] ,
      • as per this table below .

    	Column 1	Column 2	Column 3	Column 4	Column 5	Column 6	Column 7	Column 8
    ********	********	********	********	********	********	********	********	********
    Row 1	1	7	11	13	17	19	23	29
    Row 2	31	37	41	43	47	49	53	59
    Row 3	61	67	71	73	77	79	83	89
    Row 4	91	97	101	103	107	109	113	119
    Row 5	121	127	131	133	137	139	143	149
    Row 6	151	157	161	163	167	169	173	179
    Row 7	181	187	191	193	197	199	203	209

• We then knock-off the 8 [ New Possibilties ] divisible by [ 7 ] ,
  • i.e. the numbers [ 7 / 49 / 77 / 91 / 119 / 133 / 161 / 203 ] ,

    to arrive at the 48 { Candidates-for-Primes } for the [ Modulo 210 Prime Number System ] .

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Setting up the preliminaries for our study below :

• Let us first set up the prelimaries ahead of a full study of the Expansion Mechanism .

  We then set up , for the [ Modulo M Prime Number System ] , the following :

  • the value [ T-sub-Q ] :
    • 

  • the set [ Set Tau ] :
    • 

      or

    • 

• Let us also rename the [ next prime-number ] responsible for the Expansion Process as the value [ E ] :
  • 

    for easier-notation purposes .

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Studying the Expansion Process :

• For our study of the Expansion Process , let us now select :
  • any one of the elements in [ Set Tau ]

    as the core sample-element for our demonstration of the Expansion Process .

  We then label the selected sample-element as [ Tau-sub-FIXED ] :

  • 

• Let us now do the expansion based on [ Tau-sub-FIXED ] :
  • by adding successive { multiples-of-[ M ] } to [ Tau-sub-FIXED ] .

  We then have the following [ New Possibilites ] for the { candidate-for-primes } for the [ new modulo ] :

  • 

  • 

  • 

  • ............... ,
  • ............... ,

  • 

  • 

  And in general , the expansion formula here is :

  • 

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The next Question to Answer here :

• The next question we need to answer here is :
  • Are any of these { NP-sub-[ i ] }'s divisible by [ E ] ?

  To answer this question , let us now set up the { gamma-sub-[ i ] }'s given by :

  • 

    or

  • 

• Let us also set up the values [ K1 ] and [ K2 ] based on { Tau-sub-FIXED } :
  • 

    and

  • 

  And we take special notice here that [ K2 ] is never [ zero ] because :

  • [ M ] and [ E ] are always relatively prime with respect to one-another .

• Let us now bring-in the last equation from above :
  • 

  and substitute therein the values [ K1 ] and [ K2 ] to arrive at this equation :

  • 

  And since the value of [ i ] runs from [ 1 ] to [ E ] but never exceeds [ E ] ,

  • we can now come to the conclusion that :
    • 

    based on the equation immediately above .

• Let us now summarize our findings so far :
  • the range for the { Gamma-sub-[ i ] }'s runs from [ 0 ] to [ E - 1 ] ,
    • this being the regular range for [ modulo E ] ;

  • the Total Number of { Gamma-sub-[ i ] }'s is [ E ] ;

  • the { Gamma-sub-[ i ] }'s are all differently-valued and no 2 { Gamma-sub-[ i ] }'s can share the same value .

  This then lead us necessarily to conclude that :

  • one the { Gamma-sub-[ i ] }'s is valued at [ 0 ] ,

    and

  • only one of the { Gamma-sub-[ i ] }'s is valued at [ 0 ] .

  And that-is-to-say :

  • 

  • 

  This then leads us to conclude that :

  • only one of the { NP-sub-[ i ] }'s is divisible by [ E ] ,

    and

  • the remaining [ E - 1 ] { NP-sub-[ i ] }'s are not divisible by [ E ] .

• Since the above process is applicable to each-and-every element of the set [ Set Tau ] ,
  • we can now write :
    • 

  And that-is-to-say :

  • When we move from any given [ Modulo M ] to the next-higher [ Modulo M ]
    • the number of { Candidates-for-Primes } in the higher [ Modulo M Prime Number System ] is given by :
      • the number of { Candidates-for-Primes } in the lower [ Modulo M Prime Number System ]

        multiplied by :

      • a factor equal to [ E - 1 ] ;

      with the value of [ E ] here being the value of the prime-number taking responsibility

      • for lifting the lower [ Modulo M ] to the higher [ Modulo M ] .

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The Formula for the { Candiates-for-Primes } :

• Based on the equation above , we can now make this statement :

  The Statement :

  • For any { Modulo M Prime Number System } where the value of [ M ] is a { Product-of-Primes } ,
    • i.e. :
      • 

    the number of { candidates-for-primes } for that particular [ modulo M ] is given by :

    • the value { phi-of-[ Q ] }:
      • 
• Rather interesting , I would say .

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go to the next Section : Section AT-4 - A Second Look at the { Frequency-of-Primes }

go to the last Section : Section AT-2 - The { Frequency-of-Primes } for [ Modulo M ]

Return to the : Structural Integrity of Prime Number Systems HomePage

Original dated 2012-10-23