On Counting Natural Numbers up to [ N = 1543 ]

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

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Section III - Explaining the Choice of [ N = 1543 ]

Summary for the Section :

• In this Section , we shall explain our choice of :
  • setting the Upper Limit of our Count at [ N = 1543 ] .

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The Criteria for our Choice :

• For our demonstration purposes here , our first consideration here is that :
  • the Upper Limit for our count should be set sufficiently large ,

    so that :

    • the number of elements in each of [ Category II / III / IV / V ] would be numerous enough

      to give us a fairly good first-impression on the overall distribution pattern for all categories .

• Our first thought here is therefore to select a number somewhere in-between :
  • [ 1024 ] being the smallest number in [ Category X ]

    and

  • [ 2048 ] being the smallest number in [ Category XI ] .

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The Choice of [ N = 1543 ] :

• We simply note here that the first 2 numbers in [ Catgegory X ] are :
  • [ 1024 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

  • [ 1536 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

  And our choice here is simply the next [ prime number ] greater than [ 1536 ] ,

  • which happens to be [ 1543 ] .

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The Choice of a Prime Number :

• There is a specific reason why we have chosen a [ prime number ] as the Upper Limit .

  And the reason here is that :

  • the [ square-root ] , [ cube-root ] , [ 4th-root ] etc. of a [ prime number ] is never a [ whole number ] .

  As such :

  • when we use the [ square-root ] / [ cube-root ] / etc. of [ N ]
    • as the [ dividing-line / separator ] of prime-numbers / composite-numbers in later Sections ,

    the said [ dividing-line / separator ] will also never fall on a [ whole number ] .

  This then give as the distinct advantage of :

  • not having to ponder what-to-do

    if the [ dividing-line /separator ] were to hit a [ whole number ]

    • under specific circumstances .

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go to the next section : Section IV - The Powers-of-[ 2 ] in relation to [ N = 1543 ]

go to the last section : Section II - Defining the Categories for our Counting Purposes

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08