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 coordinates of the modulo
residues with respect to each prime.
For instance the
infinite coordinates for room 4 can be expressed thus:
2n, 3n+1, 5n + 4,
7n + 4 and pn + 4 for every prime thereafter.
The coordinates
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 nonzero 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 coordinates 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 nonzero residue corridor in every dimension and create an
admissible set of coordinates.
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 coordinate. (Though we need to be aware
that powers of primes also only have one zero residue coordinate, 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 coordinates 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 multidimensional
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
coordinates. 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 colourcoded 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
coprime 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 allcomposite 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.
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