On the Match-Up of Structural Patterns within Prime Number Systems

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

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Section IV - Summarizing data on the [ Modulo 30 Prime Number System ]

Summary for the Section :

• In this Section , we shall do a summary of the data for the 3rd paper in the series :
  • On Counting { Candidates-for-Primes Modulo 30 } up to [ N = 1543 ] ,
    • on a { by-category-and-by-type } basis .

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The Summary on a { by-category-and-by-type } basis :

• We then have this [ First Table ] below :
  • for the Summary Count of all { candidates-for-primes } in the [ Modulo 30 Prime Number System ] up to [ N = 1543 ] ,
    • on a { by-category-and-by-type } basis .

  Category of Number		Actual Count by Type				Roving Count	As Percent of TOTAL
  		Type A	Type B	Type C
  *************	*	*************	*************	*************	*	**********	********
  Category Zero						1	0.24
  Category I						240	58.25
  Category II		45	112			157	38.11
  Category III		4	9	1		14	3.40
  *************	*	*************	*************	*************	*	**********	********
  TOTAL						412	100.00

• And we take note here that there are a total of 171 composite-numbers here :
  • in { Category II } and { Category III } .

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Split-Percentage for the [ 171 Composite Numbers ] :

• We then have this [ Second Table ] below :
  • showing the split-percentages for the [ 171 Composite-Numbers ] .

  Category of Number		Actual Count by Type				Roving Count	As Percent of TOTAL
  		Type A	Type B	Type C
  *************	*	*************	*************	*************	*	**********	********
  Category II		45	112			157	91.81
  Category III		4	9	1		14	8.19
  *************	*	*************	*************	*************	*	**********	********
  Column Total		49	121	1		171	100.00
  Percentage Split		28.65	70.76	0.58		100.00

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Calculation of the [ Full Fields ] by Category and by Type :

• We take note at this point that :
  • [ 1531 ] is the next prime-number smaller than [ N = 1543 ] ,
    • and [ 1531 ] is the 242th Prime Number ;

  • [ 7 ] is the smallest prime-number under consideration in the [ Modulo 30 Prime Number System ] ,
    • and [ 7 ] is the 4th Prime Number .

  There are therefore :

  • 239 prime-numbers running from [ 7 ] thru [ 1531 ] ,
    • with both [ 7 ] and [ 1531 ] being inclusive thereof .

• Let us take a quick look as to how these 239 prime-numbers can be split-up :
  • on a Category-by-Category basis ,
    • based on the power-roots of [ N = 1543 ] .

  Values for the [ i-th root ] of [ N ]		* * *	Modulo 30 Prime Number System
  Description	Value of i-th -root ( X )		Number of Prime Numbers smaller than [ X ] but greater than or equal to [ 7 ]	Number of Prime Numbers greater than [ X ] but smaller than [ 1543 ]	*	Split Ratio	*	Applicable Category
  SELF	1543.0000	***	239	0		239 : 0		Category I
  Square-Root	39.2810	***	9	230		9 : 230		Category II
  Cube-Root	11.5555	***	2	237		2 : 237		Category III
  4th-Root	6.2675	***	0	239		0 : 239		Category IV
  ***********	***********	***	************************	************************	*	*************	*	*************

• We then have this [ Third Table ] below ,
  • showing the { Full Fields by Category and by Type }
    • based on the above [ Split Ratios ] of the { 239 prime-numbers } .

  Category of number		Full Fields by Type based on Split Ratio				Split Ratio
  		Type A	Type B	Type C
  *************	*	*************	*************	*************	*	**********
  Category II		45	2070			9 : 230
  Category III		4	690	53130		2 : 237
  *************	*	*************	*************	*************	*	**********

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The { Hit-Percentage Table } :

• We then have this [ Fourth Table ] below ,
  • showing the [ Hit-Percentage ] on a by Catgeory and by Type basis ,

    with the values here being :

    • [ values in the Second Table ] divided by corresponding [ values in the Third Table ] .

  Category of Number		Hit Percentage by Type				Split Ratio
  		Type A	Type B	Type C
  *************	*	*************	*************	*************	*	**********
  Category II		100.00	5.41			9 : 230
  Category III		100.00	1.30	0.0019		2 : 237
  *************	*	*************	*************	*************	*	**********

• This then concludes our summary on the { Modulo 30 Prime Number System } for the time-being .

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go to the next section : Section V - Setting up the { CHI-Axis } for our Graphics Analysis

go to the last section : Section III - Summarizing data on the [ All-Natural Prime Number System ]

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08