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 V - Counting { Category II Candidates } up to [ N ]

Summary for the Section :

• In this Section , we shall go thru the actual counting process for { Category II 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 our consideration here .

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

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 7 x [ 220.43 ]

      involving [ N ] , [ 7 ] , and their quotient [ 220.43 ] .

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

  • that :
    • [ 211 ] is the { next [ prime number ] smaller than [ 220.43 ] } , and
    • [ 223 ] is the { next [ prime number ] greater than [ 220.43 ] } .

• Consequently , we can now write :
  • [ 1477 ] = 7 x [ 211 ]
  • [ 1561 ] = 7 x [ 223 ]

  And we take note here that :

  • 1477 < 1543 < 1561

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 211 ] .

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Setting the Dividing Lines :

• Let us recall that the square-root of [ 1543 ] is roughly [ 39.28 ] .

  We can then establish :

  • [ 37 ] is the next [ prime number ] smaller than [ 39.28 ] , and
  • [ 41 ] is the next [ prime number ] greater than [ 39.28 ] .

  Consequently , we can now sub-divide the { Category II candidates } in-question here into two (2) sub-groups :

  • Type A :
    • These then are { [ Category II candidates ] less-than-or-equal-to [ N = 1543 ] } with :
      • both prime-factors being less-than-or-equal-to [ 37 ] .

  • Type B :
    • These then are { [ Category II candidates ] less-than-or-equal-to [ N = 1543 ] } with :
      • one prime-factor being less-than-or-equal-to [ 37 ] , and
      • the other prime-factor being greater-than-or-equal-to [ 41 ] .

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The Counting Procedure for { Category II Candidates } :

• Let us now do the counting for both [ Type A ] and [ Type B ] { Category II candidates } in-one-shot ,
  • and this is done as a matter of convenience .

  We then have this table below for the actual count :

  Prime Number to be investigated		* * *	Value of [ N ] divided by [ P-i ]	* * *	Next Prime smaller than [ Z ]		* * *	Next Prime greater than [ Z ]		* * *	Roving Count T = j - i + 1
  Value of Prime Number	Rank of Prime Number				Value of Prime Number	Rank of Prime Number		Value of Prime Number	Rank of Prime Number
  ( P-i )	( i )	*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( T )
  7	4	***	220.43	***	211	47	***	223	48	***	44
  11	5	***	140.27	***	139	34	***	149	35	***	30
  13	6	***	118.69	***	113	30	***	127	31	***	25
  17	7	***	90.76	***	89	24	***	97	25	***	18
  19	8	***	81.21	***	79	22	***	83	23	***	15
  23	9	***	67.09	***	67	19	***	73	21	***	11
  29	10	***	53.21	***	53	16	***	59	17	***	7
  31	11	***	49.77	***	47	15	***	53	16	***	5
  37	12	***	41.70	***	41	13	***	43	14	***	2
  **********	**********	***	***********	***	**********	**********	***	**********	**********	***	*******
  TOTAL											157

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Total Count for Category II :

• The Total Count of { Category II candidates } up to [ N = 1543 ] is therefore [ 157 ] .

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

go to the last section : Section IV - The Power-Roots of [ N ]

return to the Counting via [ modulo 30 ] HomePage

Original dated 2012-10-22