An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

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Section VII - A Fast-Track Approach to identify { prime numbers }

Summary for the Section :

• In this Section , we shall outline a fast-track procedure to identify { prime numbers } , from-the-bottom-up , i.e. :
  • we shall start with the { prime numbers } [ 2 ] and [ 3 ] .

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Using [ modulo 6 ] :

• We note here that [ 2 ] and [ 3 ] are the first two (2) { prime numbers } , so that :
  • 2 x 3 = 6

  As such [ 2 mod 6 ] , [ 3 mod 6 ] , [ 4 mod 6 ] and [ 6 mod 6 ] are { non-prime numbers } ,

  • being divisible by either [ 2 ] or [ 3 ] , or both ;

  so that :

  • [ 1 mod 6 ] and [ 5 mod 6 ] are our { candidates for prime numbers } in the next stage .

• In fact , [ 5 ] is a { prime number } .

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Using [ modulo 30 ] :

• We note here that [ 2 ] , [ 3 ] , and [ 5 ] are the first three (3) { prime numbers } , so that :
  • 2 x 3 x 5 = 30

  And [ 5 ] is then our { pivotal prime number } in this interim stage .

• Let us now list-out the additional { candidates for prime numbers } up to [ 30 ] , based on :
  • [ 1 mod 6 ] and [ 5 mod 6 ] above ;
    • as per this table below :

    Range	Candidates for Prime Numbers up to [ 30 ]		Comments
    **********	************	************	************
    1 thru 6	1	5	base row
    7 thru 12	7	11	candidate row
    13 thru 18	13	17	candidate row
    19 thru 24	19	23	candidate row
    25 thru 30	25	29	candidate row

• We note here that :
  • 25 = 5 x 5 , and this is a { non-prime number } ,

  so that our { prime numbers } here in-between [ 6 ] to [ 30 ] are then :

  • { 7 , 11 , 13 , 17 , 19 , 23 , 29 } .

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Using [ modulo 210 ] :

• We note here that [ 2 ] , [ 3 ] , [ 5 ] and [ 7 ] are the first four (4) { prime numbers } , so that :
  • 2 x 3 x 5 x 7 = 210

  And [ 7 ] is our { pivotal prime number } in this interim stage .

• Let us now list-out the additional { candidates for prime numbers } up to [ 210 ] , based on :
  • { [ 1 , 7 , 11 , 13 , 17 , 19 , 23 and 29 ] mod 30 } .
    • as per this table below :

    Range	Candidates for Prime Numbers up to [ 210 ]								Comments
    **********	****	****	****	****	****	****	****	****	************
    1 thru 30	1	7	11	13	17	19	23	29	base row
    31 thru 60	31	37	41	43	47	49	53	59	candidate row
    61 thru 90	61	67	71	73	77	79	83	89	candidate row
    91 thru 120	91	97	101	103	107	109	113	119	candidate row
    121 thru 150	121	127	131	133	137	139	143	149	candidate row
    151 thru 180	151	157	161	163	167	169	173	179	candidate row
    181 thru 210	181	187	191	193	197	199	203	209	candidate row

• We note here that there are 48 candidate elements in this table above , ( last 6 rows x 8 elements in each row ) :
  • 36 of these candidate elements here are in fact { prime numbers } , as marked-off in { pale-white color } ;

  • the remaining 12 candidate elements are then { non-prime numbers } , as marked-off { green / pale-green color } .

  And we can use this 2-Dimensional { 7 x 7 multiplication table } below :

  • to knock-off the { non-prime numbers } ;

    	7	11	13	17	19	23	29
    7	49	77	91	119	133	161	203
    11	77	121	143	187	209	253	319
    13	91	143	169	221	247	299	377
    17	119	187	221	289	323	391	493
    19	133	209	247	323	361	437	551
    23	161	253	299	391	437	529	667
    29	203	319	377	493	551	667	841
    ****	****	****	****	****	****	****	****

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Using [ modulo 2310 ] :

• We note here that [ 2 ] , [ 3 ] , [ 5 ] , [ 7 ] and [ 11 ] are the first five (5) { prime numbers } , so that :
  • 2 x 3 x 5 x 7 x 11 = 2310

  And [ 11 ] is then our { pivotal prime number } in this interim stage .

• But the { base row } for [ modulo 210 ] here is necessarily different from previously .

  This time , it is made-up of :

  • The single element [ 1 ] , intrinsic and indispensible , as always ;

  • the 42 { prime numbers } in-between [ 11 ] and [ 210 ] , same scheme as before ;

  • { composite numbers } from these { 42 prime numbers } that are less than [ 210 ] , i.e. the numbers :
    • 121 = 11 x 11 ,
    • 143 = 11 x 13 ,
    • 169 = 13 x 13 ,
    • 187 = 11 x 17 , and
    • 209 = 11 x 19 .

      And these 5 numbers are all not divisible by [ 2 ] , [ 3 ] , [ 5 ] or [ 7 ] .

    The { base row } will then have [ 1 + 42 + 5 ] , or 48 elements .

• Let us now construct the { candidate elements table } , as shown below :
  • so that there are now 480 new cadidate elements here . ( last 10 rows x 48 elements in each row )

Candidates Table

• And we were able to knock-off :
  • 178 { non-prime numbers } here using a 2-Dimensional [ 42 x 42 multiplication table ] ;
    ( the elements knocked-off here are marked-off in { purple / dark-purple color } in the table above ) .

  • 5 { non-prime numbers } here using a 3-Dimensional [ 42 x 42 x 42 multiplication table ] ;
    • ( the elements knocked-off here are marked-off in { light-pink / magenta color } in the table above ) .

  so that :

  • the remaining 297 candidate elements are in fact { prime numbers } .

• The 2-Dimensional [ 42 x 42 multiplication table ] is then as shown in Appendix B ,
  • but we have not actually constructed the 3 Dimensional [ 42 x 42 x 42 multiplication table ] here .

  We simply note here that the 5 { non-prime numbers } knocked-off via the said 3-D { multiplication table } are :

  • 1331 = 11 x 11 x 11 ,
  • 1573 = 11 x 11 x 13 ,
  • 1859 = 11 x 13 x 13 ,
  • 2057 = 11 x 11 x 17 , and
  • 2157 = 13 x 13 x 13 .

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Summarizing the progress so far :

• Let us now summarize the progess thus far :
  • The first 3 { prime numbers } are [ 2 ] , [ 3 ] , and [ 5 ] ;

  • There are 7 { prime numbers } in-between [ 7 ] and [ 30 ] ;

  • There are 36 { prime numbers } in-between [ 31 ] and [ 210 ] ;

  • There are 297 { prime numbers } in-between [ 211 ] and [ 2310 ] .

  The [ hit-rate ] for the very-last session was :

  • 297 primes out of a possible 480 candiates , a roughly 62 per cent success-rate .

• We shall take a closer look at the { multiplication tables } , next .

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go to the next section : Section VIII - Area Under the Hyperbola

go to the last section : Section VI - Actual Statistics for [ M = 31 ]

return to the Prime Numbers / L.C.M. / Zeta Functions HomePage

original dated 2009-5-08