On the Unique-Factorization Equation within Prime Number Systems

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

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Section XII - Short-Cut #3 - Using [ modulo 210 ] in lieu

Summary for the Section :

• In this Section , we shall elaborate on the 3rd short-cut , namely :
  • via using a [ higher-order modulo Prime Number System ] ,

    i.e. :

    • the [ Modulo 210 Prime Number System ] .

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The Basics of the { Moudlo 210 Prime Number System } :

• Let us recall that :
  • [ 210 ] = 2 x 3 x 5 x 7

  so that :

  • the smallest prime-number under consideration in the { Modulo 210 Prime Number System } is the prime-number [ 11 ] .

• We than have this table below ,
  • listing the { 48 candidates-for-primes modulo 210 } .

  Candidates-for-Primes modulo 210
  1	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67
  71	73	79	83	89	97	101	103	107	109	113	121	127	131	137	139
  143	149	151	157	163	167	169	173	179	181	187	191	193	197	199	209
  *****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****

  ( Hint :

  • we can use an expansion procedure on { 1 / 7 / 11 / 13 / 17 / 19 / 23 / 29 modulo 30 }

    to arrive at the 48 { candidates-for-primes modulo 210 } here . )

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{ Candidates-for-Primes modulo 210 } up to [ N = 1543 ] :

• We then have this table below ,
  • listing the 354 { candidates-for-primes modulo 210 } up to [ 1543 ] .

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

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The [ Start-Off Position ] :

• And based on the above table ,
  • we now have this diagram below for the [ Start-Off Position ] .

  

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Applying Short-Cut #1 :

• Let us now bring-in [ Short-Cut #1 ] and we have :
  • [ 121 ] = [ 11 x 11 ] .

  

  And { candidates-for-primes } less than [ 121 ] are now marked-off as [ prime-numbers ] .

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Applying Short-Cut #2 :

• Let us now bring-in Short-Cut #2 ] and we have :
  • [ 1543 ] divided by [ 11 ] = [ 140.27 ]

  

  And since we have established that [ 121 ] is not a prime-number ,

  • this then leaves us only 4 { candidates-for-primes } to work with ,

    i.e. :

    • [ 127 ] / [ 131 ] / [ 137 ] / [ 139 ] .

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What's next :

• This then prompted us to look at , next :
  • whether or not each successive [ product-of-primes ] can be used as the segregators on the [ Number-Line ] .

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go to the next section : Section XIII - { Product-of-Primes } as the segregators on the [ Number-Line ]

go to the last section : Section XI - Short-Cut #2 - [ 1543 over 7 ]

return to the Unique-Factorization Equation Homepage

Original dated 2012-10-08