
Unexpected Beauty in Primes - MrXOR
https://medium.com/cantors-paradise/unexpected-beauty-in-primes-b347fe0511b2
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hateful
Since first seeing the Ulam spiral on Numberphile, it wasn't surprising to me
that lines like that would appear. Being that prime numbers are all the
combinations by which you cannot stack a two dimensional area of pixels in a
fully filled rectangle.

e.g.

... ... ...

9 is not prime, because you can fill all rows evenly.

... .. ..

7 is, because you can't fill all the rows evenly. So the gaps make lines...

Or to put it another way, the lines in the Ulam spiral are the pixels next to
the corner pixels that make full rectangles that can't be made full rectangles
another way.

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excessive
The first time I saw these, I thought it hinted at some deep insight into the
nature of primes. However, after watching the following video, I think it just
says something about modular arithmetic and rational approximations of 2 Pi.

    
    
        https://www.youtube.com/watch?v=EK32jo7i5LQ
    

The video is only about Archimedian spirals, but I suspect a similar analysis
would apply to Ulam spirals too. We really like to find patterns.

~~~
ramshorns
The comparison to a spiral with random points plotted doesn't seem to be
enough to show that this is a pattern in the primes. It would be interesting
to see, say, a spiral with everything that's not a multiple of 2 or 3 plotted.
Maybe the spiral of primes just looks like it has a pattern because it's a
subset of that.

~~~
excessive
Yes, that's very much what the video goes into. For any rational approximation
of 2 pi, say 22/7, you won't get lines at multiples of 2 or multiples of 11.

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Igelau
The Ulam Spiral does not show a pattern. There's nothing predictive about it.
It's just a structure that shows how some of the primes group.

They're also anything but random as the author suggests. We've known that much
since the 1st century.

They're cool to look at, but there's nothing that earth shattering here.

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craftinator
What about quasi-primes

