On the Unique-Factorization Equation within Prime Number Systems

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

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Section XIII - { Product-of-Primes } as the Segregators on the [ Number Line ]

Summary for the Section :

• In this Section , we shall do a proposal .
  • First ,
    • we propose segregating the [ Number Line ] into sections via the [ Product-of-Primes ] .

  • Second ,
    • for a given value [ N ] , we propose :
      • using the next lower-valued [ Product-of-Prime ] less than [ N ]

        as the value for [ M ] in our { Modulo M Prime Number System }

        • in our efforts to determine the { number of prime numbers up to [ N ] } .

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The Core Mechanics for the { Modulo M Prime Number System } :

• Let us bring-in at this point :
  • the first 11 prime-numbers from [ 2 ] thru [ 31 ] .

  We then present this table below ,

  • detailing the { Modulo M Prime Number System } we propose to use for each range of numbers .

  Value of [ P ]	Product of Primes up to [ P ]	* * *	Modulo M Prime Number System
  			Characteristics of Prime Number System			Applicable Range for Prime Number System			Highest Category considerations
  			Value of Modulo ( M )	Smallest Prime in P-N-S		Lower Limit	Upper Limit		based on Lower Limit	based on Upper Limit
  ********	**************	*	**************	********		**************	**************		********	********
  2	2	*
  		*	2	3		2	6		0.6309	1.6309
  3	6	*
  		*	6	5		6	30		1.1133	2.1133
  5	30	*
  		*	30	7		30	210		1.7479	2.7479
  7	210	*
  		*	210	11		210	2310		2.2299	3.2299
  11	2310	*
  		*	2310	13		2310	3 0030		3.0196	4.0196
  13	3 0030	*
  		*	3 0030	17		3 0030	51 0510		3.6390	4.6390
  17	51 0510	*
  		*	51 0510	19		51 0510	969 9690		4.4637	5.4637
  19	969 9690	*
  		*	969 9690	23		969 9690	2 23092870		5.1308	6.1308
  23	2 2309 2870	*
  		*	2 2309 2870	29		2 2309 2870	64 6969 3230		5.7088	6.7088
  29	64 6969 3230	*
  		*	64 6969 3230	31		64 6969 3230	2005 6049 0130		6.5785	7.5785
  31	2005 6049 0130	*

• Let us now take a quick look at the largest { Modulo M Prime Number System } in the table .

  For the { Modulo [ 64 6969 3230 ] Prime Number System } , the characteristics are :

  • the value for [ M ] is [ 64 6969 3230 ] ,

  • the smallest prime-number under consideration is [ 31 ] , and

  • the applicable range is from [ 64 6969 3230 ] to [ 2005 6049 0130 ] .

• Let us also take special note of the number [ 7.5785 ] ,
  • as marked-off in [ magenta-color ] on the-very-right .

  This number arise from :

  • [ 7.5785 ] is equal to :
    • the { logarithm of [ 2005 6049 0130 ] }

      divided by :

    • the { logarithm of [ 31 ] } .

  In other word :

  • { [ 31 ]-raised-to-the-[ 7.5785 power ] } is equal to [ 2005 6049 0130 ] .

• And what this tell us here is that :
  • when we use the { modulo [ 64 69693230 ] Prime Number System } in this range ,
    • the highest category of { composite numbers } that we need to deal with is { Catgegory VII } .

  The smallest { Catgegory VIII Composite Number } at :

  • [ 8258 89103 7441 ] = { [ 31 ] raised-to-the-[ 8th power ] }

    will have already exceeded the Upper Limit of [ 2005 6049 0130 ] .

• Isn't that interesting .

  But more on this later when we move back to the Title Paper :

  • An Approach to the Structural Integrity of Prime Number Systems

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go to the next section : Section XIV - Concluding Remarks

go to the last section : Section XII - Short-Cut #3 - Using [ modulo 210 ] in lieu

return to the Unique-Factorization Equation Homepage

Original dated 2012-10-08