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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Epilog I - Using { L.C.M.-based modulo's } in the Fast-Track Approach

Summary for this Epilog :

• In our Fast-Track Approach to identify { prime numbers } , first presented in Section VII :
  • we used the { Product-Of-Primes series } as the series for our { modulo's } .

  In this Epilog , we shall be using , instead , an { L.C.M.-based series } for the { modulo's } ;

  • and these { modulo's } are defined by :
    • 

      recalling , of course :

    • 

• This then demonstrates the flexibilty we have in the Fast-Track Approach ,
  • notwithstanding the possibility that other series may also be usable here .

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A Brief Comparison of the 2 Series :

• A brief comparison of the 2 series , the { Product-Of-Primes series } vs. the { L.C.M.-based series } ,
  • is then as shown in this table below :

    Prime Number		Value of the modulo for our use
    Index (i)	Actual Prime Number ( p-i )	Product-Of-Primes Series	L.C.M.-Based Series
    ******	********	*****************	*****************
    1	2	2	2
    2	3	6	6
    3	5	30	60
    4	7	210	420
    5	11	2310	2 7720
    6	13	3 0030	36 0360
    7	17	51 0510	1225 2240
    8	19	969 9690	2 3279 2560
    9	23	2 2309 2870	53 5422 8880
    10	29	64 6969 3230	1552 7263 7520
    11	31	2005 6049 0130	72 2017 7644 6800

• And we shall now proceed to do the { L.C.M.-based series } modulo's , next .

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Using [ modulo 6 ] :

• Same as in Section VII , 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 60 ] :

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

  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 [ 60 ] , based on :
  • [ 1 mod 6 ] and [ 5 mod 6 ] above ;
    • as per this table below :

    Range	Candiates for Prime Numbers up to [ 60 ]		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
    31 thru 36	31	35	candidate row
    37 thru 42	37	41	candidate row
    43 thru 48	43	47	candidate row
    49 thru 54	49	53	candidate row
    55 thru 60	55	59	candidate row

• We note here that :
  • 25 = 5 x 5 ,
  • 35 = 5 x 7 ,
  • 49 = 7 x 7 , and
  • 55 = 5 x 11 ;

    and these are { non-prime numbers } ;

  so that our { prime numbers } here in-between [ 6 ] to [ 60 ] are then :

  • { 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 } .

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Using [ modulo 420 ] :

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

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

• Let us recall that we need to add-back , into the [ base row ] , { non-prime numbers } in-between [ 7 ] and [ 60 ] that are not divisible by [ 2 ] , [ 3 ] and [ 5 ] ;
  • and to this extent , we add back the number [ 49 ] , being the only number fitting the description here .

  Let us now list-out the additional { candidates for prime numbers } up to [ 420 ] , based on :

  • { [ 1 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 49 , 53 , 59 ] mod 60 } .
    • as per this table below :

    Range	Candiates for Prime Numbers up to [ 420 ]																Comments
    **********	****	****	****	****	****	****	****	****	****	****	****	****	****	****	****	****	************
    1 thru 60	1	7	11	13	17	19	23	29	31	37	41	43	47	49	53	59	base row
    61 thru 120	61	67	71	73	77	79	83	89	91	97	101	103	107	109	113	119	candidate row
    121 thru 180	121	127	131	133	137	139	143	149	151	157	161	163	167	169	173	179	candidate row
    181 thru 240	181	187	191	193	197	199	203	209	211	217	221	223	227	229	233	239	candidate row
    241 thru 300	241	247	251	253	257	259	263	269	271	277	281	283	287	289	293	299	candidate row
    301 thru 360	301	307	311	313	317	319	323	329	331	337	341	343	347	349	353	359	candidate row
    361 thru 420	361	367	371	373	377	379	383	389	391	397	401	403	407	409	413	419	candidate row

• We note here that there are 96 candidate elements in this table above , ( last 6 rows x 16 elemnents in each row ) :
  • 64 of these cadidate elements here are in fact { prime numbers } , as marked-off in { pale-white color } ;

  • the remaining 32 candidate elements are then { non-prime numbers } , as marked-off { green / pale-green / purple color } .

  And we can use this 2-Dimensional { 14 x 14 multiplication table } below :

  • to knock-off 31 { non-prime numbers } ;

    	7	11	13	17	19	23	29	31	37	41	43	47	53	59
    7	49	77	91	119	133	161	203	217	259	287	301	329	371	413	* *
    11	77	121	143	187	209	253	319	341	407	451	473	517	583	649	* *
    13	91	143	169	221	247	299	377	403	481	533	559	611	689	767	* *
    17	119	187	221	289	323	391	493	527	629	697	731	799	901	1003	* *
    19	133	209	247	323	361	437	551	589	703	779	817	893	1007	1121	* *
    23	161	253	299	391	437	529	667	713	851	943	989	1081	1219	1357	* *
    29	203	319	377	493	551	667	841	899	1073	1189	1247	1363	1537	1711	* *
    31	217	341	403	527	589	713	899	961	1147	1271	1333	1457	1643	1829	* *
    37	259	407	481	629	703	851	1073	1147	1369	1517	1591	1739	1961	2183	* *
    41	287	451	533	697	779	943	1189	1271	1517	1681	1763	1927	2173	2419	* *
    43	301	473	559	731	817	989	1247	1333	1591	1763	1849	2021	2279	2537	* *
    47	329	517	611	799	893	1081	1363	1457	1739	1927	2021	2209	2491	2773	* *
    53	371	583	689	901	1007	1219	1537	1643	1961	2173	2279	2491	2809	3127	* *
    59	413	649	767	1003	1121	1357	1711	1829	2183	2419	2537	2773	3127	3481	* *
    ***	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****

  And the remaining single { non-prime number } is then [ 343 ] = 7 x 7 x 7 ,

  • and this can be knocked-off using a { 3-D multiplication table } .

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Using [ Modulo 27720 ] :

• We simply note here that the next { modulo } in-line is [ 27720 ] , this being :
  • [ 27720 ] = 2 x 3 x 11 x [ 420 ]

    with :

    • the [ 2 ] arising from the power-of-prime [ 8 ] ;
    • the [ 3 ] arising from the power-of-prime [ 9 ] ;
    • the [ 11 ] being the [ pivotal prime number ] , next-in-line after the previous [ 7 ] .

• The procedure here and throughout will remain the same ,
  • and further details are therefore not provided here .

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go to next epilog : Epilog II - The Separability of Riemann's f(x) Function into its component parts

go to the last section : Section XVIII - Concluding Remarks

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

original dated 2009-5-17