## 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)