On Prime Number Systems and Core Structural Pattern Propagations

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

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Section IV - Other [ Prime Number Systems ] under our consideration

Summary for the Section :

• In this Section , we shall take a quick look at the possibility of other [ Prime Number Systems ] ,
  • which may be useful in our quest for a better understanding of [ prime numbers ] .

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[ Prime Number Systems ] using different modulo's :

• Let us recall that in the [ Modulo 30 Prime Number System ] :
  • the number [ 30 ] is the product of the first 3 [ prime numbers ] ,

    i.e. :

    • [ 30 ] = 2 x 3 x 5

  Our proposal here then is that :

  • we can always come-up with a { Modulo-M Prime Number System } ,
    • where the value [ M ] is the { product of the prime-numbers up to [ N ] } .

• We then have a partial listing for the possible values of [ M ] ,
  • as per the middle column in [ light-purple color ] in the table below .

    Value of [ N ]	* *	Product of Prime Numbers up to [ N ]	* *	L.C.M. of Natural Numbers up to [ N ]
    ********	*	****************	*	****************
    2		2		2
    3		6		6
    5		30		60
    7		210		420
    11		2310		27720
    13		3 0030		36 0360
    17		51 0510		1225 2240
    19		969 9690		2 3279 2560
    23		2 2309 2870		53 5422 8880
    29		64 6969 3230		2 3290 8956 2800

• We also take special note here of an alternative and that is :
  • For the value of [ M ] in the { Modulo-M Prime Number System } ,
    • we can also use the L.C.M. of the { natural numbers from [ 2 ] to [ N ] } .

  We then have a partial listing for the possible L.C.M.'s ,

  • as per the right-hand-side column in [ light-green color ] in the table above .

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Setting up the { Modulo-M Prime Number System } :

• The procedure for setting up a { Modulo-M Prime Number System } is then as follows :

  STEP ONE :

  • Once we have chosen an appropriate value for [ M ] ,
    • we can then identify the [ values modulo M ] that are divisible by the { prime-factors of [ M ] } .

  STEP TWO :

  • We then set up the redundant set of [ values modulo M ] that are not divisible by the { prime-factors of [ M ] } .

  STEP THREE :

  • We then propose the redundant set of [ values modulo M ] as the { candidates-for-primes }
    • for the { Modulo-M Prime Number System } .

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go to the next section : Section V - The Reasons for using different modulo's for the [ Prime Number Systems ]

go to the last section : Section III - The [ Modulo 30 Prime Number System ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08