Twin Prime Proof Appendix

Continued from Page 3

Appendix

Here’s a bit more detail on how the first hypercuboid was constructed – this is the simple version before we added the complication of residues of squares. This is a version that is looking for pn and pn+2 numbers (eg twin primes) starting from the point where we have narrowed the possible numbers down to the 2n+1 planes and the 3n+1 planes:

1	7	13	19	25
31	37	43	49	55
61	67	73	79	85
91	97	103	109	115
121	127	133	139	145
151	157	163	169	175
181	187	193	199	205

Numbers coloured grey are either 5N or 5N +2.

Numbers coloured blue are either 7N or 7N+2.

We turn this into a box by adding the following five slices starting from each of the five columns:

1	31	61	91	121	151	181
211	241	271	301	331	361	391
421	451	481	511	541	571	601
631	661	691	721	751	781	811
841	871	901	931	961	991	1021
1051	1081	1111	1141	1171	1201	1231
1261	1291	1321	1351	1381	1411	1441
1471	1501	1531	1561	1591	1621	1651
1681	1711	1741	1771	1801	1831	1861
1891	1921	1951	1981	2011	2041	2071
2101	2131	2161	2191	2221	2251	2281

7	37	67	97	127	157	187
217	247	277	307	337	367	397
427	457	487	517	547	577	607
637	667	697	727	757	787	817
847	877	907	937	967	997	1027
1057	1087	1117	1147	1177	1207	1237
1267	1297	1327	1357	1387	1417	1447
1477	1507	1537	1567	1597	1627	1657
1687	1717	1747	1777	1807	1837	1867
1897	1927	1957	1987	2017	2047	2077
2107	2137	2167	2197	2227	2257	2287

13	43	73	103	133	163	193
223	253	283	313	343	373	403
433	463	493	523	553	583	613
643	673	703	733	763	793	823
853	883	913	943	973	1003	1033
1063	1093	1123	1153	1183	1213	1243
1273	1303	1333	1363	1393	1423	1453
1483	1513	1543	1573	1603	1633	1663
1693	1723	1753	1783	1813	1843	1873
1903	1933	1963	1993	2023	2053	2083
2113	2143	2173	2203	2233	2263	2293

19	49	79	109	139	169	199
229	259	289	319	349	379	409
439	469	499	529	559	589	619
649	679	709	739	769	799	829
859	889	919	949	979	1009	1039
1069	1099	1129	1159	1189	1219	1249
1279	1309	1339	1369	1399	1429	1459
1489	1519	1549	1579	1609	1639	1669
1699	1729	1759	1789	1819	1849	1879
1909	1939	1969	1999	2029	2059	2089
2119	2149	2179	2209	2239	2269	2299

25	55	85	115	145	175	205
235	265	295	325	355	385	415
445	475	505	535	565	595	625
655	685	715	745	775	805	835
865	895	925	955	985	1015	1045
1075	1105	1135	1165	1195	1225	1255
1285	1315	1345	1375	1405	1435	1465
1495	1525	1555	1585	1615	1645	1675
1705	1735	1765	1795	1825	1855	1885
1915	1945	1975	2005	2035	2065	2095
2125	2155	2185	2215	2245	2275	2305

And as before, note that we can rearrange these to make the planes clearer, for instance:

	5n+1	5n+2	5n+3	5n+4	5n+5
7n+1	1	127	43	169	85
7n+2	121	37	163	79	205
7n+3	31	157	73	199	115
7n+4	151	67	193	109	25
7n+5	61	187	103	19	145
7n+6	181	97	13	139	55
7n+7	91	7	133	49	175

	7n+1	7n+2	7n+3	7n+4	7n+5	7n+6	7n+7
11n+1	1	331	661	991	1321	1651	1981
11n+2	211	541	871	1201	1531	1861	2191
11n+3	421	751	1081	1411	1741	2071	91
11n+4	631	961	1291	1621	1951	2281	301
11n+5	841	1171	1501	1831	2161	181	511
11n+6	1051	1381	1711	2041	61	391	721
11n+7	1261	1591	1921	2251	271	601	931
11n+8	1471	1701	2131	151	481	811	1141
11n+9	1681	2011	31	361	691	1021	1351
11n+10	1891	2211	241	571	901	1231	1561
11n+11	2101	121	451	781	1111	1441	1771

	7n+1	7n+2	7n+3	7n+4	7n+5	7n+6	7n+7
11n+1	463	793	1123	1453	1783	2113	133
11n+2	673	1003	1333	1663	1993	13	343
11n+3	883	1213	1543	1873	2203	223	553
11n+4	1093	1423	1753	2083	103	433	763
11n+5	1303	1633	1963	2293	313	643	973
11n+6	1513	1843	2173	193	523	853	1183
11n+7	1723	2053	73	403	733	1063	1393
11n+8	1933	2263	283	613	943	1273	1603
11n+9	2143	163	493	823	1153	1483	1813
11n+10	43	373	703	1033	1363	1693	2023
11n+11	253	583	913	1243	1573	1903	2233

For the proof above, these diagrams would need to be extended to 25n x 49n grids and 49n x 121n grids, but it is easier just to imagine those…

Appendix B

Just a quick thought on modulo residues:


I find it easiest to think of modulo residues this way – any prime number p has a modulo residue of p for every other prime. For instance, 19 is defined by the infinite set of pn + 19 residues. However, for numbers below 19, you need to convert this into numbers in the appropriate modulo range, so this is equivalent to 2n+1, 3n+1, 5n+4, 7n+5, 11n+8, 13n+6, 17n+2, 19n (or 19n+19) and every pn+19 thereafter. 

Appendix C

Some people might not like the "merging" stage of the attempted proof. (it's certainly a bit dodgy as things stand!) An alternative method would be to simply remove the pn+2 planes from the grid at the same time as repeating other green planes. This would leave us with an incomplete number chain, with repetitions. It would still be topologically equivalent to the prime number hypercuboid. And when it comes to the p^2n+p and p^2n+p+2 planes, we could simply ignore one of them since both will contain the same numbers (>3) in locations where they are not intersected by black planes.

(See also this page for an attempt to simplify the patterns of coloured planes)

Appendix D

Sometimes in the proof there may be some confusion between planes and hyperplanes - I've not been too pedantic about this because the only truly important thing about the multi-dimensional geometry of the hypercuboid is that each number lies on the intersection of an infinite number of planes and that an (admissible) infinite range of modulo residues uniquely defines any number.