New Twin Prime Proof Attempt (flawed, but slightly interesting...)

I’ve been working on this topological twin prime proof for a while. It isn't quite there, but I think it is an interesting near miss, and potentially could be improved. So I'm putting it up for the time being, with a few comments (in red) on where the problems are.

The hypergeometry of twin primes

Overview

1. Multiples of prime numbers can be arranged as planes in a hypercuboid and sorted by modulo residue, with the planes intersecting at right angles.
2. Numbers that are not multiples of particular prime numbers can also be arranged in the same way.
3. The number chain can thus be arranged into a hypercuboid shape resembling a multi-dimensional Menger Sponge. (Alternatively, since it is a Hilbert space it can be imagined as a multi-dimensional version of Hilbert’s Hotel.)
4. The locations of rooms in the hotel can be seen as infinite sets of co-ordinates defining particular numbers.
5. There is a topological equivalence between the geometric patterns within the hypercuboid for primes and twin primes. We can show this by “stretching” the original hypercuboid into a form in which the only remaining primes are the highest of two twin primes (this is one of the two dodgy bits - see the comments later on).
6. With a bit of further analysis, this method proves that there is an infinitude of twin primes.

1. Multiples of particular prime numbers can be seen as intersecting planes in a

cuboid.

Take the first six numbers and arrange them in a 2 x 3 rectangle.

1	3	5
2	4	6

Next, colour in multiples of 2.

1	3	5
2	4	6

Then we take the number line up to 30 (the primorial 2 x 3 x 5 = 5#)
 

26	28	30
20	22	24
14	16	18
8	10	12
2	4	6
2	4	6
1	3	5
1	3	5
7	9	11
13	15	17
19	21	23
25	27	29

Imagine this folded into a box, with the first six numbers on top (central yellow section), and four more layers of 6 beneath.

(I've repeated 1-6 to emphasize this 3-D view) but won't do the same in future grids.)

We colour in the numbers that are multiples of 3. 

At this stage we have a cuboid, 2x3x5, in which there is a coloured plane (even numbers) and a coloured column.

Next, imagine a line of 7 cuboids, laid out this way, containing the numbers up to 210, you can see that the 2n and 2n+1 numbers form planes. Here are the first three:

 

26	28	30		56	58	60		86	88	90
20	22	24		50	52	54		80	82	84
14	16	18		44	46	48		74	76	78
8	10	12		38	40	42		68	70	72
2	4	6		32	34	36		62	64	66
1	3	5		31	33	35		61	63	65
7	9	11		37	39	41		67	69	71
13	15	17		43	45	47		73	75	77
19	21	23		49	51	53		79	81	83
25	27	29		55	57	59		85	87	89

The odd 3n numbers form a plane that intersects with the 2n numbers:

The 5n numbers (yellow) form lines – if you take only the odd numbers from this image then add a line of boxes below 210 higher, they will form planes (technically hyperplanes as we are going into a fourth dimension) intersecting at a right angle to the remaining 3n plane. (See next section for clarification as to why this can be seen as a single plane).

 

1	3	5		31	33	35		61	63	65
7	9	11		37	39	41		67	69	71
13	15	17		43	45	47		73	75	77
19	21	23		49	51	53		79	81	83
25	27	29		55	57	59		85	87	89
211	213	215		241	243	245		271	273	275
217	219	221		247	249	251		277	279	281
223	225	227		253	255	257		283	285	287
229	231	233		259	261	263		289	291	293
235	237	239		265	267	269		295	297	299

These extend into infinite planes.

In the diagrams above all numbers in the coloured planes are composite, with the exceptions of 2, 3, and 5.

To clarify this, we can rearrange the original diagrams thus:

 

	3n+1	3n+2	3n
	10	20	30	5n	40	50	60		70	80	90
	4	14	24	5n+4	34	44	54		64	74	84
2n	28	8	18	5n+3	58	38	48		88	68	78
	22	2	12	5n+2	52	32	42		82	62	72
	16	26	6	5n+1	46	56	36		76	86	66
	1	11	21	5n+1	31	41	51		61	71	81
	7	17	27	5n+2	37	47	57		67	77	87
2n+1	13	23	3	5n+3	43	53	33		73	83	63
	19	29	9	5n+4	49	59	39		79	89	69
	25	5	15	5n	55	35	45		85	65	75

This can be seen more clearly as a depiction of three planes at right angles. The 5n plane is horizontal, and at a right angle to the page and includes the even 5n numbers which aren’t marked.

(It might be more helpful to imagine each box containing an arithmetic progression, so rather than multiple grids, the 11 square in the first grid contains the arithmetic progression 11 + 30n, and so on).

2. Numbers that are not multiples of prime numbers can also be seen in the same way.

If we take the diagrams above and colour the numbers that are not multiples of 2, 3 and 5, they also form intersecting planes.

	3n+1	3n+2	3n+3
	10	20	30
	28	26	24
2n + 2	22	14	18
	16	8	12
	4	2	6
	1	11	21
	7	17	27
2n+1	13	23	3
	19	29	9
	25	5	15

 Anywhere these planes meet, the number at the meeting point cannot be a multiple of 2, 3, or 5. For instance at 19, 23 or 49, planes of numbers that are not multiples of 2, 3 or 5 all intersect.

This process continues for higher primes. The diagrams get a bit cumbersome, but here are a couple of sections showing how part of the 5n plane and 7n plane looks:

	5n+1	5n+2	5n+3	5n+4	5n
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	91	7	133	49	175

And here is that same section extended into the 11n dimension.

	7n+1	7n+2	7n+3	7n+4	7n+5	7n+6	7n
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	2101	121	451	781	1111	1441	1771

What we are constructing is a multidimensional hypercuboid – each extra prime number we consider takes us to a higher dimensional cuboid. (Incidentally, it doesn’t matter what order the numbers are “added” in because the final analysis relies on a hypercuboid of infinite dimensions in which each number naturally has a certain position, not on the way we have to rearrange the number chain in order to understand this pattern).

So the entire number chain can be arranged as an infinite-dimensional hypercuboid with modulo residue planes that intersect at right angles to each other.

(Please note, I'm not claiming that the method above proves that there is an infinitude of primes - it is just a way of visualising the number chain within a hypercuboid - we already know there is an infinitude of primes, so we know something about how the planes in this hypercuboid must intersect).

I’ll give some more detail on the construction of this hypercuboid in Appendix A as these last two steps are a bit briefly explained. However I’d rather go on with the main argument for now.

Go to Page 2 (there are 3 pages and an appendix in total)