On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section XII - Formulating { Category X Numbers } with 8 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 8 prime-numbers ;

    with the 2 special conditions being that :

    • each of the 8 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 8 Prime-Numbers A-B-C-D-E-F-G-H				* * *	Roving Count
  **************		**************	**************	**************	**************		**************
  3-1-1-1-1-1-1-1		3-1-1-1-1-1-1-1	1-3-1-1-1-1-1-1	1-1-3-1-1-1-1-1	1-1-1-3-1-1-1-1		8
  		1-1-1-1-3-1-1-1	1-1-1-1-1-3-1-1	1-1-1-1-1-1-3-1	1-1-1-1-1-1-1-3
  2-2-1-1-1-1-1-1		2-2-1-1-1-1-1-1	2-1-2-1-1-1-1-1	2-1-1-2-1-1-1-1	2-1-1-1-2-1-1-1		28
  		2-1-1-1-1-2-1-1	2-1-1-1-1-1-2-1	2-1-1-1-1-1-1-2	1-2-2-1-1-1-1-1
  		1-2-1-2-1-1-1-1	1-2-1-1-2-1-1-1	1-2-1-1-1-2-1-1	1-2-1-1-1-1-2-1
  		1-2-1-1-1-1-1-2	1-1-2-2-1-1-1-1	1-1-2-1-2-1-1-1	1-1-2-1-1-2-1-1
  		1-1-2-1-1-1-2-1	1-1-2-1-1-1-1-2	1-1-1-2-2-1-1-1	1-1-1-2-1-2-1-1
  		1-1-1-1-1-1-2-1	1-1-1-2-1-1-1-2	1-1-1-1-2-2-1-1	1-1-1-1-2-1-2-1
  		1-1-1-1-2-1-1-2	1-1-1-1-1-2-2-1	1-1-1-1-1-2-1-2	1-1-1-1-1-1-2-2
  **************		**************	**************	**************	**************		**************
  		TOTAL					36

• A quick explanation on the { Nominal Format } on the left-hand-most column :
  • for the 1st entry of [ 3-1-1-1-1-1-1-1 ] , 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 ,
      • 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 ,
      • the 6th prime-number is raised to the 1st-power ,
      • the 7th prime-number is raised to the 1st-power , and
      • the 8th 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 [ 8 Prime-Numbers ] :

• The Total Count for originally starting-out with 8 prime-numbers is therefore :
  • [ 36 ]

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go to the next section : Section XIII - Formulating [ Category X Numbers ] with 9 prime-numbers

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

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08