## Sunday, 15 July 2012

### Twin Prime Proof page 2

Continued from page 1

3. The hypercuboid hotel

So let’s imagine the entire number chain laid out as a hotel in the shape of this hypercuboid.

From one angle it looks like this.

 2n 2n+1 2 1 4 3 6 5 8 7 10 9

where n is zero or any positive integer * (You could imagine this as the start of two infinite series of  numbered rooms - odd numbers and even numbers– you walk through room 1 to get to room 3, 5, 7 etc.)

Turned through a right angle it looks like this

 3n 3n+1 3n+2 3 1 2 6 4 5 9 7 8 12 10 11

Turned through another right angle it looks like this

 5n 5n+1 5n+2 5n+3 5n+4

Each dimension is laid out like this for a higher prime, with the layers arranged by modulo residue. Within the 5n plane, all the rooms are composite, except for the first one, number 5. The numbers in the other four corridors are not multiples of 5.

Within the hotel, we can define any number by an infinite set of co-ordinates of the modulo residues with respect to each prime.

For instance the infinite co-ordinates for room 4 can be expressed thus:
2n, 3n+1, 5n + 4, 7n + 4 and pn + 4 for every prime thereafter.

The co-ordinates for room 5 can be expressed thus:
2n+1, 3n+2, 5n, 7n + 5 and pn + 5 for every prime thereafter.

How do we “find” prime numbers in the hotel?

A first naïve thought is that you can simply “walk down” a non-zero residue corridor in every dimension. For instance you could walk down the pn + 1 corridor on every plane. This actually does create the set of co-ordinates for the number 1, as 1 has a modulo residue of 1 for every other number. However this is the only way you can choose a non-zero residue corridor in every dimension and create an admissible set of co-ordinates.
This is because every number has a zero residue in its own modulo plane, for instance, in the 5n plane, the number 5 has a zero residue even though it is a prime number. 35 has a zero residue for 5 and 7 and so on.

If a number is prime, it has only one zero residue co-ordinate. (Though we need to be aware that powers of primes also only have one zero residue co-ordinate, for instance 25 and 125 only have a zero residue at 5n - this is a problem we will deal with later in this proof.)

If a number has two or more zero residue co-ordinates it is definitely not a prime.

4. The hypercuboid and the Menger Sponge

At this point, the imaginary hypercuboid can be seen to resemble a multi-dimensional Menger Sponge, the fractal object which can be compared in turn to a Sierpinski Carpet taken up a dimension. All we are doing here is extending that analysis to higher dimensions. Just to clarify this – imagine a slightly different way of visualizing the hotel.

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

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

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

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

We can arrange the layers of the hotel so that in each dimension we are taking a symmetrical “slice” through the centre. Of course this doesn’t produce a symmetrical pattern in the number chain, as we have had to rearrange the number chain to fit this visualization. But we can see more clearly how the construction of this hypercuboid is similar to the construction of a Menger Sponge or a Cantor Set.

It is accepted that, taken to infinity, the Menger Sponge approaches zero volume but still has an infinite surface area.

However the geometry here is slightly different – there aren’t any “holes” because every number is on its own modulo zero plane. So we need to think slightly differently about the geometry of the hypercuboid.

When we look for primes, we are looking for the points that are only intersected by one zero residue plane. The more zero residue planes a point is intersected by, the more different factors that number has.

Next, in order to deal with the problem of powers of primes it is necessary to use a slightly more complicated version of the hypercuboid. Then we can (finally) move on to the twin prime proof.

(nb C.R.Greathouse, for whom I have a lot of respect, thinks the Menger Sponge analogy is weak, which is fair enough - certainly on its own it doesn't prove anything though I don't think anything else that follows depends on it, it's just a way I thought of visualising the hypercuboid).

5. The hypercuboid arranged by squares of primes

In order to make sure we can’t confuse primes with powers of primes we are going to rearrange the hypercuboid so that in each dimension we are considering the modulo residues of squares of primes, instead of primes.

For instance, rather than dividing the numbers into 2n and 2n+1, we divide them into 4n, 4n+1, 4n+2, 4n+3.

2 is a 4n+2 number. All powers of 2 (4, 8, 16 etc) are 4n numbers. For any prime p, p is now in a different plane to p^2 (p squared), p^3 etc. In general, p will be in the (p^2)n+p plane, and not in any other pn plane.

Within this hypercuboid each number still has a unique set of co-ordinates. We can now say without exception that a prime number is one that is only intersected by one (p^2)n+p plane.

Here is a colour-coded visual representation of this, for residues of 4 and 9.

 4n 4n+1 4n+2 4n+3 9n 36 9 18 27 9n+1 28 1 10 19 9n+2 20 29 2 11 9n+3 12 21 30 3 9n+4 4 13 22 31 9n+5 32 5 14 23 9n+6 24 33 6 15 9n+7 16 25 34 7 9n+8 8 17 26 35

·         Green = numbers that are co-prime to 2 and 3
·         Red = p^2n+p = prime, if only intersected by one red plane across all dimensions, otherwise not prime (for instance in the grid above, 2 and 3 are red but won't be intersected by any other red planes)
·         Black = all composite numbers
·         Blue = intersection of two red planes, therefore a composite.
·
When we go up to the next dimension, we will have 25 slices coloured thus:
·         Black: 25n, 25n+10, 25n+15, 25n+20
·         Coloured the same as above: All other planes except for 25n+5
·         The 25n + 5 plane will be coloured blue where it coincides with the red squares above, and red where it coincides with green squares above (as in the image below).

 4n 4n+1 4n+2 4n+3 9n 180 405 630 855 9n+1 280 505 730 55 9n+2 380 605 830 155 9n+3 480 705 30 255 9n+4 580 805 130 355 9n+5 680 5 230 455 9n+6 780 105 330 555 9n+7 880 205 430 655 9n+8 80 305 530 755

5 is the only remaining number on this section that will not be intersected by another red plane (and turned blue) in other dimensions.

Since we are marking numbers as composite if they are on the intersection of two p^2n+p (red) planes, and we are not mixing up powers of primes with primes, we are not missing any composite numbers in this sieving process.
All numbers will end up coloured red, blue or black. If we imagine each number as a room in the infinite hotel, they all start off green, then get painted black if an all-composite plane intersects them, red if an (p^2)n+p plane intersects them, and blue if two or more (p^2)n+p planes intersect them. Numbers that remain red must be primes.

We haven't at this point proved anything about prime distribution. However we know (from Euclid and other prime proofs) that there remains an infinitude of numbers in the hypercuboid that are only intersected by one (red) (p^2)n+p plane, because we know there is an infinitude of primes.

So far this has just been an elaborate way of visualising something we already knew about, but the interesting thing is how we can extend this analysis to twin primes.