On the Unique-Factorization Equation within Prime Number Systems

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

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Section III - Setting up the [ 412 Empty Bins ]

Summary for the Section :

• In this Section , we shall set up a set of { 412 empty bins } ,
  • in preparation for our combinatorics analysis on the { Modulo 30 Prime Number System } ,
    • in the 5 Sections to follow .

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Recalling the 412 { Candidates-for-Primes } for the [ Modulo 30 Prime Number System ] :

• Let us recall that :
  • the { candidates-for-primes } for the [ Modulo 30 Prime Number System ] are :
    • { 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 modulo 30 } .

  We then have this table below ,

  • listing-out the 412 { candidates-for-primes } for the [ Modulo 30 Prime Number System ] 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 a Special Note here :
  • The above table is the same as the one we have in Section III of the 3rd Paper in the current series ,
    • i.e. the paper :

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

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Setting up the [ 412 Empty Bins ] :

• We then set up 412 empty bins ,
  • with each bin being mapped to a { candidate-for-prime } in the above table

    and

    marked-off accordingly with the numerals for that particular { candidate-for-prime } .

  We then have this schematic presentation of the 412 empty bins ,

  • as per this diagram below .

  

• And this set of { 412 empty bins } shall now be collectively known as the [ 412 Empty Bins ] ,
  • and shall be refered to as such in our combinatorics analysis to follow .

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go to the next section : Section IV - The a-priori Assumptions and the [ Start-Off Position ]

go to the last section : Section II - A Statement of Objectives

return to the Unique-Factorization Equation HomePage

Original dated 2012-10-08