On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section VI - Formulating { Category X Numbers } with 2 prime-numbers

Summary for the Section :

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

  QUESTION :

  • How many distinct-and-different { Category X Composite Numbers } can we formulate
    • if we were to start-out with exactly 2 prime-numbers ;

    with the 2 special conditions being that :

    • each of the 2 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 2 Prime-Numbers A-B		* * *	Roving Count
  *********	*	***********	***********	*	*********
  9-1		9-1	1-9		2
  8-2		8-2	2-8		2
  7-3		7-3	3-7		2
  6-4		6-4	4-6		2
  5-5		5-5			1
  *********	*	**********	**********	*	*********
  		TOTAL			9

• A quick explanation on the { Nominal Format } on the left-hand-most column :
  • for the 1st entry of [ 9-1 ] , this means that :
    • for the 2 prime-numbers being arranged in any order :
      • the 1st prime-number is raised to the 9th-power , and
      • the 2nd 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 [ 2 Prime-Numbers ] :

• The Total Count for originally starting-out with 2 prime-numbers is therefore :
  • [ 9 ]

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go to the next section : Section VII - Formulating [ Category X Numbers ] with 3 prime-numbers

go to the last section : Section V - Formulating [ Category X Numbers ] with 1 prime-number

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08