On Unique-Factorization Combinatorics via the Pascal Triangle

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

Top Of Page

Section VIII - Formulating { Category X Numbers } with 4 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 4 prime-numbers ;

    with the 2 special conditions being that :

    • each of the 4 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 .

go to Top Of Page

Finding the Answer :

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

  Nominal Format	* * *	Listing of the Powers for the 4 Prime-Numbers A-B-C-D						* * *	Roving Count
  *********	*	********	********	********	********	********	********	*	*********
  7-1-1-1		7-1-1-1	1-7-1-1	1-1-7-1	1-1-1-7				4
  6-2-1-1		6-2-1-1	6-1-2-1	6-1-1-2	2-6-1-1	1-6-2-1	1-6-1-2		12
  		2-1-6-1	1-2-6-1	1-1-6-2	2-1-1-6	1-2-1-6	1-1-2-6
  5-3-1-1		5-3-1-1	5-1-3-1	5-1-1-3	3-5-1-1	1-5-3-1	1-5-1-3		12
  		3-1-5-1	1-3-5-1	1-1-5-3	3-1-1-5	1-3-1-5	1-1-3-5
  5-2-2-1		5-2-2-1	5-2-1-2	5-1-2-2	2-5-2-1	2-5-1-2	1-5-2-2		12
  		2-2-5-1	2-1-5-1	1-2-5-2	2-2-1-5	2-1-2-5	1-2-2-5
  4-4-1-1		4-4-1-1	4-1-4-1	4-1-1-4	1-4-4-1	1-4-1-4	1-1-4-4		6
  4-3-2-1		4-3-2-1	4-3-1-2	4-2-3-1	4-2-1-3	4-1-3-2	4-1-2-3		24
  		3-4-2-1	3-4-1-2	2-4-3-1	2-4-1-3	1-4-3-2	1-4-2-3
  		3-2-4-1	3-1-4-2	2-3-4-1	1-3-4-2	2-1-4-3	1-2-4-3
  		3-2-1-4	3-1-2-4	2-3-1-4	1-3-2-4	2-1-3-4	1-2-3-4
  4-2-2-2		4-2-2-2	2-4-2-2	2-2-4-2	2-2-2-4				4
  3-3-3-1		3-3-3-1	3-3-1-3	3-1-3-3	1-3-3-3				4
  3-3-2-2		3-3-2-2	3-2-3-2	3-2-2-3	2-3-3-2	2-3-2-3	2-2-3-3		6
  *********	*	********	********	********	********	********	********	*	*********
  		TOTAL							84

• A quick explanation on the { Nominal Format } on the left-hand-most column :
  • for the 1st entry of [ 7-1-1-1 ] , this means that :
    • for the 3 prime-numbers being arranged in any order :
      • the 1st prime-number is raised to the 7th-power ,
      • the 2nd prime-number is raised to the 1st-power ,
      • the 3rd prime-number is raised to the 1st-power , and
      • the 4th 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 .

go to Top Of Page

Total Count for [ 4 Prime-Numbers ] :

• The Total Count for originally starting-out with 84 prime-numbers is therefore :
  • [ 84 ]

go to Top Of Page

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

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

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08