On Counting [ Candidates-for-Primes modulo 30 ] up to [ N = 1543 ]

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

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Section VII - Counting { Category III Candidates } up to [ N ]

Summary for the Section :

• In this Section , we shall go thru the actual counting process for { Category III Candidates } up to [ N ] ,
  • with [ N ] here being the natural number [ 1543 ] .

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Setting-up the Upper-Limit for our investigations here :

• Let us recall that [ 7 ] is the smallest [ prime number ] under consideration here .

  And , for our purpose of counting { Category III candidates } here ,

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 7 x 7 x [ 31.49 ]

      involving [ N ] , the { square-of-[ 7 ] } , and their quotient [ 31.49 ] .

  We can then quickly establish , from our [ Table of Prime Numbers ] ,

  • that :
    • [ 31 ] is the { next [ prime number ] smaller than [ 31.49 ] } , and
    • [ 37 ] is the { next [ prime number ] greater than [ 31.49 ] } .

• Consequently , we can now write :
  • [ 1519 ] = 7 x 7 x [ 31 ]

  • [ 1813 ] = 7 x 7 x [ 37 ]

  And we take note here that :

  • 1519 < 1543 < 1813

  And what this tells us here then is that :

  • it will be most unlikely that :
    • any { [ Category III candidate ] less-than-or-equal-to [ N = 1543 ] }

      will involve a [ prime factor ] greater than the prime-number [ 31 ] .

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Setting the Dividing Lines :

• Let us recall again that the cube-root of [ 1543 ] is roughly [ 11.56 ] .

  We can then establish :

  • [ 11 ] is the next [ prime number ] smaller than [ 11.56 ] , and
  • [ 13 ] is the next [ prime number ] greater than [ 11.56 ] .

  Consequently , we can now sub-divide the { Category III candidates } in-question here into three (3) sub-groups :

  • Type A :
    • These then are { [ Category III candidates ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 3 prime-factors being less-than-or-equal-to [ 11 ] .

  • Type B :
    • These then are { [ Category III candidates ] less-than-or-equal-to [ N = 1543 ] } with :
      • 2 prime-factors being less-than-or-equal-to [ 11 ] , and
      • 1 prime-factor being greater-than-or-equal-to [ 13 ] .

  • Type C :
    • These then are { [ Category III candidates ] less-than-or-equal-to [ N = 1543 ] } with :
      • 1 prime-factor being less-than-or-equal-to [ 11 ] , and
      • 2 prime-factors being greater-than-or-equal-to [ 13 ] .

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The Counting Procedure for { Category III Candidates } :

• Let us now do the counting of { Category III candidates } in two (2) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category III candidates } in one-shot ;
      • and this is done as a matter of convenience .

  • Stage II :
    • Counting [ Type C ] { Category III candidates } .

• STAGE I :

  We then set up this table below for the actual count :

  Category II Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Next Prime smaller than [ Z ]		* * *	Next Prime larger than [ Z ]		* * *	Control Value for Counting Purposes		* * *	Roving Count T = j - i + 1
  Value of Cat. II Number	Factors of Cat. II Number				Value of Prime	Rank of Prime		Value of Prime	Rank of Prime		Largest Prime Factor of [ X ]	Rank of Prime
  ( X )		*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( P-i )	( i )	*	( T )
  49	7 x 7	***	31.49	***	31	11	***	37	12	***	7	4	***	8
  77	7 x 11	***	20.04	***	19	8	***	23	9	***	11	5	***	4
  121	11 x 11	***	12.75	***	11	5	***	13	6	***	11	5	***	1
  **********	**********	***	********	***	********	********	***	********	********	***	********	********	***	********
  TOTAL														13

• STAGE II :

  We then set up this table below for the actual count .

  Category II Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Next Prime smaller than [ Z ]		* * *	Next Prime larger than [ Z ]		* * *	Roving Count T = j - 3
  Value of Cat. II Number	Factors of Cat. II Number				Value of Prime	Rank of Prime		Value of Prime	Rank of Prime
  ( X )		*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( T )
  169	13 x 13	***	9.13	***	7	4	***	11	5	***	1
  **********	**********	***	********	***	********	********	***	********	********	***	********
  TOTAL											1

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Total Count for Category III :

• The Total Count of { Category III candidates } up to [ N = 1543 ] is therefore :
  • [ 14 ] = 13 + 1

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go to the next section : Section VIII - Counting Summary for all Categories

go to the last section : Section VI - Counting { Category II Candidates } up to [ N ]

return to the Counting via [ modulo 30 ] HomePage

Original dated 2012-10-08