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# Prime factorisation

<https://en.wikipedia.org/wiki/Prime_number>

The prime factorisation of the integers is a central point of study in number theory and can be visualised with the variant of Ulam's spiral.

Ulam's spiral plots the primes in a polar coordinate system (p,p) where the second component is in radians (π/2 rad = 90°, π rad = 180°, 3π/2 = 270°, 2π rad = 360°).

We can first consider the set ℕ, plotting all (n,n) pairs, with (0,0) as the origin; the increments in the first component make each sucessive point 1 unit further from the center, and the second component is the angle in radians.

So, we have something like:

```
0°=360°|       | 90°          | 180°         | 270°                 | 360°
k*2π   |       | k*π/2        | k*π          | k*3π/2               | k*2π
(0,0)  | (1,1) | 1.57 | (2,2) | 3.14 | (3,3) | 4.71 | (5,5) | (6,6) | 6.28
  ^      ^              ^              ^              ^       ^
plotted points for pairs (n,n) where n ∈ {0..6}
```

If we overlay x-axis as the kπ axis (for k ∈ ℕ), the pair (6,6) is fairly off (relatively, much worse compared to other candidates) the x-axis (or, it is off from the  point), but nevertheless it is the last one before a full circle is completed. Therefore, there are 6 spiraling arms coming out of the origin point. Each of the 6 spiraling arm is one of the six residue classes modulo 6, `6k+n mod 6`.

| RCm6 | 0 | 1  | 2  | 3  | 4  | ... | ℙ?                                      |
| ---- | - | -- | -- | -- | -- | --- | --------------------------------------- |
| 6k+0 | 0 | 6  | 12 | 18 | 24 | ... | none, 𝔼(n) -> ¬ℙ(n) (for n > 2)        |
| 6k+1 | 1 | 7  | 13 | 19 | 25 | ... | *most*                                  |
| 6k+2 | 2 | 8  | 14 | 20 | 26 | ... | none, 𝔼 -> ¬ℙ                          |
| 6k+3 | 3 | 9  | 15 | 21 | 27 | ... | 3 ∈ ℙ, others are multiples of 3 so ∉ ℙ |
| 6k+4 | 4 | 10 | 16 | 22 | 28 | ... | none, 𝔼 -> ¬ℙ                          |
| 6k+5 | 5 | 11 | 17 | 23 | 29 | ... | *most*                                  |

Therefore, to plot the primes, we remove the non-prime arms (6k+0, 6k+2, 6k+3, 6k+4), and we are left with (most members) of the 6k+1 and 6k+5 arms; that is, there are just 2 arms, 6k+5 and 6k+7 (since 6k+1 = 6k+7).

Anyway, the points that lay really close to the x-axis are 22 and 44, that is, the pairs (22,22) and (44,44), which will turn out to be significant magic numbers later.