On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section X - Formulating { Category X Numbers } with 6 prime-numbers

Summary for the Section :

• In this Section , we shall attempt to answer the question :

  QUESTION :

  • How many distinct-and-different { Category X Composite Numbers } can we formulate
    • if we were to start-out with exactly 6 prime-numbers ;

    with the 2 special conditions being that :

    • each of the 6 prime-numbers must appear at-least-once as a prime-factor of the formulated-number [ N ] ,

      and

    • repetition of a prime-number as a prime-factor of the formulated-number [ N ] is allowed .

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Finding the Answer :

• We then have this table below :
  • to help us find the answer .

  Nominal Format	* * *	Listing of the Powers for the 6 Prime-Numbers A-B-C-D-E-F						* * *	Roving Count
  ***********	*	***********	***********	***********	***********	***********	***********	*	***********
  5-1-1-1-1-1		5-1-1-1-1-1	1-5-1-1-1-1	1-1-5-1-1-1	1-1-1-5-1-1	1-1-1-1-5-1	1-1-1-1-1-5		6
  4-2-1-1-1-1		4-2-1-1-1-1	1-4-2-1-1-1	1-1-4-2-1-1	1-1-1-4-2-1	1-1-1-1-4-2	2-1-1-1-1-4		30
  		4-1-2-1-1-1	1-4-1-2-1-1	1-1-4-1-2-1	1-1-1-4-1-2	2-1-1-1-4-1	1-2-1-1-1-4
  		4-1-1-2-1-1	1-4-1-1-2-1	1-1-4-1-1-2	2-1-1-4-1-1	1-2-1-1-4-1	1-1-2-1-1-4
  		4-1-1-1-2-1	1-4-1-1-1-2	2-1-4-1-1-1	1-2-1-4-1-1	1-1-2-1-4-1	1-1-1-2-1-4
  		4-1-1-1-1-2	2-4-1-1-1-1	1-2-4-1-1-1	1-1-2-4-1-1	1-1-1-2-4-1	1-1-1-1-2-4
  3-3-1-1-1-1		3-3-1-1-1-1	3-1-3-1-1-1	3-1-1-3-1-1	3-1-1-1-3-1	3-1-1-1-1-3	1-3-3-1-1-1		15
  		1-3-1-3-1-1	1-3-1-1-3-1	1-3-1-1-1-3	1-1-3-3-1-1	1-1-3-1-3-1	1-1-3-1-1-3
  		1-1-1-3-3-1	1-1-1-3-1-3	1-1-1-1-3-3
  3-2-2-1-1-1		3-2-2-1-1-1	1-3-2-2-1-1	1-1-3-2-2-1	1-1-1-3-2-2	2-1-1-1-3-2	2-2-1-1-1-3		30
  		3-2-1-2-1-1	1-3-2-1-2-1	1-1-3-2-1-2	2-1-1-3-2-1	1-2-1-1-3-2	2-1-2-1-1-3
  		3-2-1-1-2-1	1-3-2-1-1-2	2-1-3-2-1-1	1-2-1-3-2-1	1-1-2-1-3-2	2-1-1-2-1-3
  		3-2-1-1-1-2	2-3-2-1-1-1	1-2-3-2-1-1	1-1-2-3-2-1	1-1-1-2-3-2	2-1-1-1-2-3
  		3-1-2-2-1-1	1-3-1-2-2-1	1-1-3-1-2-2	2-1-1-3-1-2	2-2-1-1-3-1	1-2-2-1-1-3
  		3-1-2-1-2-1	1-3-1-2-1-2	2-1-3-1-2-1	1-2-1-3-1-2	2-1-2-1-3-1	1-2-1-2-1-3
  		3-1-2-1-1-2	2-3-1-2-1-1	1-2-3-1-2-1	1-1-2-3-1-2	2-1-1-2-3-1	1-2-1-1-2-3
  		3-1-1-2-2-1	1-3-1-1-2-2	2-1-3-1-1-2	2-2-1-3-1-1	1-2-2-1-3-1	1-1-2-2-1-3
  		3-1-1-2-1-2	2-3-1-1-2-1	1-2-3-1-1-2	2-1-2-3-1-1	1-2-1-2-3-1	1-1-2-1-2-3
  		3-1-1-1-2-2	2-3-1-1-1-2	2-2-3-1-1-1	1-2-2-3-1-1	1-1-2-2-3-1	1-1-1-2-2-3
  2-2-2-2-1-1		2-2-2-2-1-1	2-2-2-1-2-1	2-2-1-2-2-1	2-1-2-2-2-1	1-2-2-2-2-1	2-2-2-1-1-2		15
  		2-2-1-2-1-2	2-1-2-2-1-2	1-2-2-2-1-2	2-2-1-1-2-2	2-1-2-1-2-2	1-2-2-1-2-2
  		2-1-1-2-2-2	1-2-1-2-2-2	1-1-2-2-2-2
  ***********	*	***********	***********	***********	***********	***********	***********	*	***********
  		TOTAL							126

• A quick explanation on the { Nominal Format } on the left-hand-most column :
  • for the 1st entry of [ 5-1-1-1-1-1 ] , this means that :
    • for the 6 prime-numbers being arranged in any order :
      • the 1st prime-number is raised to the 5th-power ,
      • the 2nd prime-number is raised to the 1st-power ,
      • the 3rd prime-number is raised to the 1st-power ,
      • the 4th prime-number is raised to the 1st-power ,
      • the 5th prime-number is raised to the 1st-power , and
      • the 6th prime-number is raised to the 1st-power ;

      in the formulation of the { Category X Composite Number } .

  And this type of Format Scheme is carried throughout in the above table .

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Total Count for [ 6 Prime-Numbers ] :

• The Total Count for originally starting-out with 6 prime-numbers is therefore :
  • [ 126 ]

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go to the next section : Section XI - Formulating [ Category X Numbers ] with 7 prime-numbers

go to the last section : Section IX - Formulating [ Category X Numbers ] with 5 prime-numbers

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08