On the Unique-Factorization Equation within Prime Number Systems

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

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Section VII - An Example on Knocking-Off [ Non-Prime Numbers ]

Summary for the Section :

• In this Section , we shall do an example on { knocking-off non-prime numbers }
  • with [ 11 / 17 / 91 / 121 / 221 ] being the 5 { Assumed Prime-Numbers } here .

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Setting up the set of [ Assumed Prime-Numbers ] :

• For our example here in this Section ,
  • we shall now mark-off the following five (5) { candidates-for-primes modulo 30 } :
    • [ 11 ] , [ 17 ] , [ 91 ] , [ 121 ] and [ 221 ] ;

      as our [ Assumed Prime-Numbers ] .

  We then have this diagram below ,

  • for the associated { Input String of 412-elements } .

  

• And the set of { Assumed Prime-Numbers } that shall be taking responsibility
  • for knocking-off { non-prime candidates-for-primes }

  shall then consist of the 6 elements :

  • [ 7 ] / [ 11 ] / [ 17 ] / [ 91 ] / [ 121 ] / [ 221 ] .

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Setting up the 21 { 2-elements Combinations } :

• We then have this table below ,
  • listing-out the { 2-elements combinations } arising from the above { set of 6-elements } .

  	* *	DOUBLE		* *	SINGLE + SINGLE
  		Combination	Value		Combination	Value	*	Combination	Value	*	Combination	Value
  **	*	************	*********	*	************	********	*	************	********	*	************	********
  1	*	7 x 7	49	*	7 x 11	77	*	11 x 17	187	*	17 x 121	2057
  2	*	11 x 11	121	*	7 x 17	119	*	11 x 91	1001	*	17 x 221	3757
  3	*	17 x 17	289	*	7 x 91	637	*	11 x 121	1331	*	91 x 121	1 1011
  4	*	91 x 91	8281	*	7 x 121	847	*	11 x 221	2431	*	91 x 221	2 0111
  5	*	121 x 121	1 4641	*	7 x 221	1547	*	17 x 91	1547	*	121 x 221	2 6741
  6	*	221 x 221	4 8841	*

• And we bring-in again this general equation for the Full Count of { 2-elements Combinations } :
  • 

    where : [ H ] is the total number of [ assumed prime-numbers ] .

  Since we have a total of 6 { Assumed Prime-Numbers } , the Full Count here is then :

  • 

    yielding :

  • 

  And consequently :

  • 

• And we simply take note that :
  • we do have exactly 21 { 2-elements combinations } in the table above .

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Setting up the 56 { 3-elements Combinations } :

• We then have this table below ,
  • listing-out the { 3-elements combinations } arising from the above { set of 6-elements } .

  	* *	TRIPLE		* *	DOUBLE + SINGLE					* *	3 SINGLES
  		Combination	Value		Combination	Value	*	Combination	Value		Comb't'n	Value	*	Comb't'n	Value
  **	*	************	*********	*	************	********	*	************	********	*	**********	********	*	**********	********
  1	*	7*7*7	343	*	7*7*11	539	*	7*11*11	847	*	7*11*17	1309	*	91*121*221	243 3431
  2	*	11*11*11	1331	*	7*7*17	833	*	7*17*17	2023	*	7*11*91	7007	*	17*121*221	45 4597
  3	*	17*17*17	4913	*	7*7*91	4459	*	7*91*91	5 7967	*	7*11*121	9317	*	17*91*221	34 1887
  4	*	91*91*91	75 3571	*	7*7*121	5929	*	7*121*121	10 2487	*	7*11*221	1 7017	*	17*91*121	18 7187
  5	*	121*121*121	177 7561	*	7*7*221	1 0829	*	7*221*221	34 1887	*	7*17*91	1 0829	*	11*121*221	29 4151
  6	*	221*221*221	1079 3861	*	11*11*17	2057	*	11*17*17	3179	*	7*17*121	1 4399	*	11*91*221	12 1121
  7	*			*	11*11*91	1 1011	*	11*91*91	9 1091	*	7*17*221	2 6299	*	11*91*121	12 1121
  8	*			*	11*11*121	1 4641	*	11*121*121	16 1051	*	7*91*121	7 7077	*	11*17*221	4 1327
  9	*			*	11*11*221	2 6741	*	11*221*221	53 7251	*	7*91*221	14 0777	*	11*17*121	2 2627
  10	*			*	17*17*91	2 6299	*	17*91*91	14 0777	*	7*121*221	18 7187	*	11*17*91	1 7017
  11	*			*	17*17*121	3 4969	*	17*121*121	24 8897	*
  12	*			*	17*17*221	6 3869	*	17*221*221	83 0297	*
  13	*			*	91*91*121	100 2001	*	91*121*121	133 2331	*
  14	*			*	91*91*221	183 0101	*	91*221*221	444 4531	*
  15	*			*	121*121*221	323 5661	*	121*221*221	590 9761	*

• And we bring-in again this general equation for the Full Count of { 3-elements Combinations } :
  • 

    where : [ H ] is the total number of [ assumed prime-numbers ] .

  Since we have a total of 6 { Assumed Prime-Numbers } , the Full Count here is then :

  • 

    yielding :

  • 

  And consequently :

  • 

• And we simply take note that :
  • we do have exactly 56 { 3-elements combinations } in the table above .

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Identifying the Duplications :

• Let us now identify the places where duplications occurred :
  • Duplication arising from [ 11 x 11 = 121 ] :

    Value	1st Combination	2nd Combination
    *********	************	************
    847	7 x 121	7 x 11 x 11
    1331	11 x 121	11 x 11 x 11
    2057	17 x 121	11 x 11 x 17
    11011	91 x 121	11 x 11 x 91
    1 4641	121 x 121	11 x 11 x 121
    2 6741	121 x 221	11 x 11 x 221

  • Duplication arising from [ 7 x 221 = 17 x 91 ] :

    Value	1st Combination	2nd Combination
    *********	************	************
    1547	7 x 221	17 x 19
    1 0829	7 x 7 x 221	7 x 17 x 91
    1 7017	7 x 11 x 221	11 x 17 x 91
    2 6299	7 x 17 x 221	17 x 17 x 91
    14 0777	7 x 91 x 221	17 x 91 x 91
    18 7187	7 x 121 x 221	17 x 91 x 121
    34 1887	7 x 221 x 221	17 x 91 x 221

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The associated { Output String of 412-elements } :

• Let us now proceed to :
  • evaluate the 21 [ 2-elements combinations ] and the 56 [ 3-elements combinations ]
    • in the 2 tables above ;

    and

  • put a [ green-color ball ] into the bin associated with the [ evaluated numerical value ]
    • whenever that [ evaluated numerical value ] comes-in at below [ N = 1543 ] .

  We then have this diagram below ,

  • for the associated { Output String of 412-elements } .

  

  And we now take special note at this point of the following :

  • For [ Bin #121 ] , we have :
    • one [ red-color ball ] arising from being an [ assumed prime-number ] and

    • one [ green-color ball ] arising from [ 121 = 11 x 11 ] .

  • For [ Bin #847 ] , we have :
    • one [ green-color ball ] arising from [ 847 = 7 x 121 ] and

    • another [ green-color ball ] arising from [ 847 = 7 x 11 x 11 ] .

  • For [ Bin #1331 ] , we have :
    • one [ green-color ball ] arising from [ 1331 = 11 x 121 ] and

    • another [ green-color ball ] arising from [ 1331 = 11 x 11 x 11 ] .

• Rather simple and straight-forward !

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go to the next section : Section VIII - The { Output String of 412-elements }

go to the last section : Section VI - Knocking-Off [ Non-Prime Numbers ]

return to the Unique-Factorization Equation Homepage

Original dated 2012-10-08