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 III - The [ Candidates-for-Primes modulo 30 ] up to [ N = 1543 ]

Summary for the Section :

• In this Section , we shall make-out the full list of { candidates for Prime Numbers } up to [ N = 1543 ] ,
  • for the sake of absolute clarity .

  And we take special notice here that :

  • the choice of [ N = 1543 ] here is the same as in the previous paper :
    • On Counting Natural Numbers up to [ N = 1543 ]

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Explaining the basics for [ candidates-for-primes modulo 30 ] :

• Let us recall that the first 3 prime-numbers are :
  • [ 2 ] , [ 3 ] , and [ 5 ] respectively .

  And their product is [ 30 ] , i.e. :

  • [ 30 ] = 2 x 3 x 5

  Thus :

  • [ numbers modulo 30 ] divisible by [ 2 ] , [ 3 ] or [ 5 ] are not [ candidates-for-primes ] ,

    whereas :

  • [ numbers modulo 30 ] not divisible by [ 2 ] , [ 3 ] or [ 5 ] are [ candidates-for-primes ] .

• And in essence :
  • the 22 elements in the set :
    • [ 2 / 3 / 5 / 6 / 8 / 9 / 10 / 12 / 14 / 15 / 16 / 18 / 20 / 21 / 22 / 24 / 25 / 26 / 27 / 28 / 30 modulo 30 ]

      are divisible by either [ 2 ] , [ 3 ] or [ 5 ] ;

    whereas :

  • the 8 elements in the redundant set :
    • [ 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 modulo 30 ]

      are not divisble by [ 2 ] , [ 3 ] or [ 5 ] .

• Therefore , the [ candidates-for-primes ] on this basis are then :
  • [ 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 modulo 30 ] .

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Listing-out the actual { Candidates for Prime Numbers } :

• We then have this table below for the full listing of the { Candidates for Primes } up to [ N = 1543 ] :

  		Candidates for Prime Numbers
  Row #1		1	7	11	13	17	19	23	29		31	37	41	43	47	49	53	59
  Row #2		61	67	71	73	77	79	83	89		91	97	101	103	107	109	113	119
  Row #3		121	127	131	133	137	139	143	149		151	157	161	163	167	169	173	179
  Row #4		181	187	191	193	197	199	203	209		211	217	221	223	227	229	233	239
  Row #5		241	247	251	253	257	259	263	269		271	277	281	283	287	289	293	299
  Row #6		301	307	311	313	317	319	323	329		331	337	341	343	347	349	353	359
  Row #7		361	367	371	373	377	379	383	389		391	397	401	403	407	409	413	419
  Row #8		421	427	431	433	437	439	443	449		451	457	461	463	467	469	473	479
  Row #9		481	487	491	493	497	499	503	509		511	517	521	523	527	529	533	539
  Row #10		541	547	551	553	557	559	563	569		571	577	581	583	587	589	593	599
  ********	*	*****	*****	*****	*****	*****	*****	*****	*****	*	*****	*****	*****	*****	*****	*****	*****	*****
  Row #11		601	607	611	613	617	619	623	629		631	637	641	643	647	649	653	659
  Row #12		661	667	671	673	677	679	683	689		691	697	701	703	707	709	713	719
  Row #13		721	727	731	733	737	739	743	749		751	757	761	763	767	769	773	779
  Row #14		781	787	791	793	797	799	803	809		811	817	821	823	827	829	833	839
  Row #15		841	847	851	853	857	859	863	869		871	877	881	883	887	889	893	899
  Row #16		901	907	911	913	917	919	923	929		931	937	941	943	947	949	953	959
  Row #17		961	967	971	973	977	979	983	989		991	997	1001	1003	1007	1009	1013	1019
  Row #18		1021	1027	1031	1033	1037	1039	1043	1049		1051	1057	1061	1063	1067	1069	1073	1079
  Row #19		1081	1087	1091	1093	1097	1099	1103	1109		1111	1117	1121	1123	1127	1129	1133	1139
  Row #20		1141	1147	1151	1153	1157	1159	1163	1169		1171	1177	1181	1183	1187	1189	1193	1199
  ********	*	*****	*****	*****	*****	*****	*****	*****	*****	*	*****	*****	*****	*****	*****	*****	*****	*****
  Row #21		1201	1207	1211	1213	1217	1219	1223	1229		1231	1237	1241	1243	1247	1249	1253	1259
  Row #22		1261	1267	1271	1273	1277	1279	1283	1289		1291	1297	1301	1303	1307	1309	1313	1319
  Row #23		1321	1327	1331	1333	1337	1339	1343	1349		1351	1357	1361	1363	1367	1369	1373	1379
  Row #24		1381	1387	1391	1393	1397	1399	1403	1409		1411	1417	1421	1423	1427	1429	1433	1439
  Row #25		1441	1447	1451	1453	1457	1459	1463	1469		1471	1477	1481	1483	1487	1489	1493	1499
  Row #26		1501	1507	1511	1513	1517	1519	1523	1529		1531	1537	1541	1543

  And the total number of { Candidates for Prime Numbers } listed above is then [ 412 ] :

  • [ 412 ] = [ 25 rows ] x [ 16 in each row ] + 12

    inclusive of both [ 1 ] and [ 1543 ] .

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The smallest Prime Number we need to consider here :

• Since :
  • all our { Candidates for Prime Numbers } here are not divisible by [ 2 ] , [ 3 ] , or [ 5 ] ;

  then :

  • { Candidates for Prime Numbers } that turn-out later to be non-prime numbers
    • must necessarily be divisible by prime/s other than [ 2 ] , [ 3 ] and [ 5 ] .

• We shall therefore start our investigations in the sections-to-come on knocking-out [ non-prime numbers ]
  • with the prime-number [ 7 ] as the first-possible [ prime factor ] for testing
    • as to whether the [ candidate-for-prime ] in-question is indeed prime or non-prime .

  And we take note here , of course , that :

  • [ 7 ] is the next smallest prime-number after [ 2 ] , [ 3 ] and [ 5 ] .

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go to the next section : Section IV - The Powers-of-[ 7 ] in relation to [ N ]

go to the last section : Section II - Defining the Categories

return to the Counting via [ Modulo 30 ] HomePage

Original dated 2012-10-08