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 (Ο/2 rad = 90Β°, Ο rad = 180Β°, 3Ο/2 = 270Β°, 2Ο rad = 360Β°).
We can first consider the set β, 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:
If we overlay x-axis as the kΟ axis (for k β β), 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
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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.
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