On Prime Number Systems and Core Structural Pattern Propagations

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

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Section III - The [ Modulo 30 Prime Number System ]

Summary for the Section :

• In this Section , we shall take a first-and-initial look at the [ Modulo 30 Prime Number System ] .

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The basis for the [ Modulo 30 Prime Number System ] :

• Let us recall that the first 3 prime numbers are [ 2 ] , [ 3 ] and [ 5 ] , respectively .

  And the product of these 3 prime numbers is [ 30 ] ,

  • i.e. :
    • [ 30 ] = 2 x 3 x 5

• Thus , for [ modulo 30 ] ,
  • we can now identify the following set of 22 numbers [ modulo 30 ] as being always divisible by either [ 2 ] , [ 3 ] or [ 5 ] :

    *************	*************	*************	*************	*************	*************
    2 modulo 30	3 modulo 30	4 modulo 30	5 modulo 30	6 modulo 30	8 modulo 30
    9 modulo 30	10 modulo 30	12 modulo 30	14 modulo 30	15 modulo 30	16 modulo 30
    18 modulo 30	20 modulo 30	21 modulo 30	22 modulo 30	24 modulo 30	25 modulo 30
    26 modulo 30	27 modulo 30	28 modulo 30	30 modulo 30

  and as a result ,

  • we can now identify the following redundant set of 8 numbers [ modulo 30 ] as being not divisible by either [ 2 ] , [ 3 ] or [ 5 ] :

    *************	*************	*************	*************
    1 modulo 30	7 modulo 30	11 modulo 30	13 modulo 30
    17 modulo 30	19 modulo 30	23 modulo 30	29 modulo 30

• Our proposal here is therefore that :
  • { 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 modulo 30 } numbers be taken as the { Candidates-for-Primes }
    • in the [ Modulo 30 Prime Number System ] .

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The [ Modulo 30 Prime Number System ] :

• In general , for the [ Modulo 30 Prime Number System ] :
  • { 1 / 7 / 11 / 13 / 17 /19 / 23 / 29 modulo 30 } numbers are proposed as [ candidates-for-primes ] .

  Subsequently , each [ candidate-for-prime ] is checked , in an ascending order ,

  • as to whether-or-not it is indeed a [ composite number ] .

  And

  • IF :
    • the particular [ candidate-for-prime ] turns-out to be a [ composite number ] and therefore a [ non-prime number ] ,

    THEN :

    • that particular [ candidate-for-prime ] is knocked-off the list
      • so that only genuine [ prime numbers ] shall remain on the [ list-of-prime-numbers ] .

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The Core Mechanics for the [ Modulo 30 Prime Number System ] :

• For the [ Modulo 30 Prime Number System ] , we shall take as [ a priori ] that :
  • [ 1 ] is the [ Super-Prime ] ,

    and

  • [ 7 ] is the first-and-smallest [ prime number ] under consideration here .

• And ,
  • since [ 7 ] is the smallest [ prime number ] under consideration here within the system ,

    it then follows that :

    • [ 49 ] = 7 x 7

      is necessarily the first-and-smallest [ composite number ] under consideration within the system ,

      • for the specific purpose of knocking-off proposed [ candidates-for-primes ] as [ non-prime numbers ] .

  As such , [ candidates-for-primes ] less than [ 49 ] may now be marked-off as [ prime numbers ] ,

  • and indeed :
    • the natural numbers [ 11 / 13 / 17 / 19 / 23 / 29 / 31 / 37 / 41 / 43 ] are [ prime numbers ] .

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go to the next section : Section IV - Other [ Prime Number Systems ] under our consideration

go to the last section : Section II - The [ All-Natural Prime Number System ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08