On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section VII - Formulating { Category X Numbers } with 3 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 3 prime-numbers ;

    with the 2 special conditions being that :

    • each of the 3 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 3 Prime-Numbers A-B-C						* * *	Roving Count
  *********	*	******	******	******	******	******	******	*	*********
  8-1-1		8-1-1	1-8-1	1-1-8					3
  7-2-1		7-2-1	7-1-2	1-7-2	2-7-1	2-1-7	1-2-7		6
  6-3-1		6-3-1	6-1-3	1-6-3	3-6-1	3-1-6	1-3-6		6
  6-2-2		6-2-2	2-6-2	2-2-6					3
  5-4-1		5-4-1	5-1-4	1-5-4	4-5-1	4-1-5	1-4-5		6
  5-3-2		5-3-2	5-2-3	2-5-3	3-5-2	3-2-5	2-3-5		6
  4-4-2		4-4-2	4-2-4	2-4-4					3
  4-3-3		4-3-3	3-4-3	3-3-4					3
  *********	*	******	******	******	******	******	******	*	*********
  		TOTAL							36

• A quick explanation on the { Nominal Format } on the left-hand-most column :
  • for the 1st entry of [ 8-1-1 ] , this means that :
    • for the 3 prime-numbers being arranged in any order :
      • the 1st prime-number is raised to the 8th-power ,
      • the 2nd prime-number is raised to the 1st-power , and
      • the 3rd 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 [ 3 Prime-Numbers ] :

• The Total Count for originally starting-out with 10 prime-numbers is therefore :
  • [ 36 ]

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go to the next section : Section VIII - Formulating [ Category X Numbers ] with 4 prime-numbers

go to the last section : Section VI - Formulating [ Category X Numbers ] with 2 prime-numbers

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08