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 III - Summarizing the data on the [ All-Natural Prime Number System ]

Summary for the Section :

• In this Section , we shall do a summary of the data for the 2nd paper in the series :
  • On Counting Natural Numbers up to [ N = 1543 ] ,
    • on a { by-category-and-by-type } basis .

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

• We then have this [ First Table ] below :
  • for the Summary Count of all { candidates-for-primes } in the [ All-Natural 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	Type D	Type E	Type F	Type G	Type H	Type J	Type K
  *************	*	*******	*******	*******	*******	*******	*******	*******	*******	*******	*******	*	**********	********
  Category Zero													1	0.06
  Category I													243	15.75
  Category II		78	372										450	29.16
  Category III		20	320	47									387	25.08
  Category IV		13	155	64	4								236	15.29
  Category V		6	62	43	11	1							123	7.97
  Category VI		7	35	16	3	0	0						61	3.95
  Category VII		1	8	7	4	3	1	1					25	1.62
  Category VIII		1	4	3	2	1	0	0	0				11	0.71
  Category IX		1	2	1	0	0	0	0	0	0			4	0.26
  Category X		1	1	0	0	0	0	0	0	0	0		2	0.13
  *************	*	*******	*******	*******	*******	*******	*******	*******	*******	*******	*******	*	**********	********
  TOTAL													1543	100.00

• And we take note here that there are 1299 composite-numbers here :
  • from { Category II } thru { Category X } .

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Split-Percentage for the [ 1299 Composite Numbers ] :

• We then have this [ Second Table ] below :
  • showing the split-percentages for the [ 1299 Composite-Numbers ] .

  Category of Number		Actual Count by Type											Roving Count	As Percent of TOTAL
  		Type A	Type B	Type C	Type D	Type E	Type F	Type G	Type H	Type J	Type K
  *************	*	*******	*******	*******	*******	*******	*******	*******	*******	*******	*******	*	**********	********
  Category II		78	372										450	34.64
  Category III		20	320	47									387	29.79
  Category IV		13	155	64	4								236	18.17
  Category V		6	62	43	11	1							123	9.47
  Category VI		7	35	16	3	0	0						61	4.70
  Category VII		1	8	7	4	3	1	1					25	1.92
  Category VIII		1	4	3	2	1	0	0	0				11	0.85
  Category IX		1	2	1	0	0	0	0	0	0			4	0.31
  Category X		1	1	0	0	0	0	0	0	0	0		2	0.15
  *************	*	*******	*******	*******	*******	*******	*******	*******	*******	*******	*******	*	**********	********
  Column Total		128	959	181	24	5	1	1	0	0	0		1299	100.00
  Percentage Split		9.85	73.83	13.93	1.85	0.38	0.08	0.08	0	0	0		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 ;

  • [ 2 ] is the smallest prime-number under consideration in the [ All-Natural Prime-Number-System ] ,
    • and [ 2 ] is the 1st Prime Number .

  There are therefore :

  • 242 prime-numbers running from [ 2 ] thru [ 1531 ] ,
    • with both [ 2 ] and [ 1531 ] being inclusive thereof .

• Let us take a quick look as to how these [ 242 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 ]		* * *	All-Natural Prime Number System
  Description	Value of i-th -root ( X )		Number of Prime Numbers smaller than [ X ] but greater than or equal to [ 2 ]	Number of Prime Numbers greater than [ X ] but smaller than [ 1543 ]	*	Split Ratio	*	Applicable Category
  SELF	1543.0000	***	242	0		242 : 0		Category I
  Square-Root	39.2810	***	12	230		12 : 230		Category II
  Cube-Root	11.5555	***	5	237		5 : 237		Category III
  4th-Root	6.2675	***	3	239		3 : 239		Category IV
  5th-Root	4.3418	***	2	240		2 : 240		Category V
  6th-Root	3.3993	***	2	240		2 : 240		Category VI
  7th-Root	2.8542	***	1	241		1 : 241		Category VII
  8th-Root	2.5035	***	1	241		1 : 241		Category VIII
  9th-Root	2.2608	***	1	241		1 : 241		Category IX
  10th-Root	2.0837	***	1	241		1 : 241		Category X
  11th-Root	1.9492	***	0	242		0 : 242		Category XI
  ***********	***********	***	************************	************************	*	*************	*	*************

• We then have this [ Third Table ] below ,
  • showing the { Full Fields by-category-and-by-Type }
    • based on the above [ Split Ratios ] of the { 242 prime-numbers } .

  CAT.		Full Fields by Type based on Split Ratio											Split Ratio
  		Type A	Type B	Type C	Type D	Type E	Type F	Type G	Type H	Type J	Type K
  *****	*	*********	*********	*********	*********	*********	*********	*********	*********	*********	*********	*	*******
  II		78	2760										12 : 230
  III		20	3555	141015									5 : 237
  IV		13	2390	172080	6911880								3 : 239
  V		6	1200	115680	6698640	283444920							2 : 240
  VI		7	1440	144600	9331520	425167380	K6-F						2 : 240
  VII		1	241	29161	2362041	144084501	K7-F	K7-G					1 : 241
  VIII		1	241	29161	2362041	144084501	K7-F	K7-G	K8-H				1 : 241
  IX		1	241	29161	2362041	144084501	K7-F	K7-G	K8-H	K9-J			1 : 241
  X		1	241	29161	2362041	144084501	K7-F	K7-G	K8-H	K9-J	K10-K		1 : 241
  *****	*	*********	*********	*********	*********	*********	*********	*********	*********	*********	*********	*	*******

  • with the values being too large to fit into the above table being as shown in this table below :

    [ K6-F ]	138 3211 2096
    [ K7-F ]	70 6014 0549
    [ K7-G ]	2894 6576 2509
    [ K8-H ]	10 2140 0619 1389
    [ K9-J ]	316 1076 9193 3059
    [ K10-K ]	8742 3701 4348 1299
    **********	**********************

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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 ] .

  CAT.		Hit Percentage by Type											Split Ratio
  		Type A	Type B	Type C	Type D	Type E	Type F	Type G	Type H	Type J	Type K
  *****	*	*********	*********	*********	*********	*********	*********	*********	*********	*********	*********	*	*******
  II		100.00	13.48										12 : 230
  III		100.00	9.00	0.0333									5 : 237
  IV		100.00	6.49	0.0372	0.000058								3 : 239
  V		100.00	5.17	0.0372	0.000157	0.00							2 : 240
  VI		100.00	2.43	0.0111	0.000032	0	0						2 : 240
  VII		100.00	3.32	0.0242	0.000171	0.00	0.00	0.00					1 : 241
  VIII		100.00	1.66	0.0104	0.000086	0.00	0	0	0				1 : 241
  IX		100.00	0.83	0.0035	0	0	0	0	0	0			1 : 241
  X		100.00	0.41	0	0	0	0	0	0	0	0		1 : 241

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

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go to the next section : Section IV - Summarizing the data on the [ Modulo 30 Prime Number System ]

go to the last section : Section II - A Quick Review of Unique-Factorization Combinatorics

return to the Matching Structural Patterns HomePage

Original dated 2012-10-08