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
 Multiples of prime numbers can be arranged as planes in a hypercuboid and sorted by modulo residue, with the planes intersecting at right angles.
 Numbers that are not multiples of particular prime numbers can also be arranged in the same way.
 The number chain can thus be arranged into a hypercuboid shape resembling a multidimensional Menger Sponge. (Alternatively, since it is a Hilbert space it can be imagined as a multidimensional version of Hilbert’s Hotel.)
 The locations of rooms in the hotel can be seen as infinite sets of coordinates defining particular numbers.
 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).
 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 16 to emphasize this 3D 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.
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 infinitedimensional 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).
(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)
Go to Page 2 (there are 3 pages and an appendix in total)
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