Sunday 15 July 2012

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)

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