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Fractal patterns in prime numbers

August 2011

I should start saying that I'm not a mathematician, just a social activist, ex-computer scientist that has visual math as a hobby, and general interest in filosophy, physics and math, specially fractals.

For several months I've been trying to represent prime numbers in different ways, invented by me, to see if I could find visual patterns hidden in them. I do that using POV (Persistance of Vision raytracer) and programming my ideas. I knew it would be basically impossible, so I just did it for fun.

I run on to the idea of painting primes in binary after several strange episodes that lead me to think I could have found some patterns in primes; but it turned out to be a POV visual error.

Using binary has the limitation of word length. I decided to use 16 bits for a word. Therefore I would only be able to represent primes smaller than 65536. To go beyond that, I would have to use 32, 64, 128... bits, and it might turn out it would only work with a determined word length.

Nevertheless, I decided to go on and paint the prime numbers that were smaller than 65.536, actually I arrived to the prime 63803, wich makes the 6.400th prime, using a 16 bit representation. I tried an increasing series of drawings, and found out what they looked like:

Here you can appreciate how I represented the first 16 prime numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Blue balls are zeros, yellow balls ones. You can see 1 (0000000000000001) in the lower row, and 47 (00000000101111) in the upper row.

Let's go on with this method:

By now you must have been able to detect a pattern in the image, and maybe you have noticed that it's not and exact pattern, but rather, a fractal pattern. Almost equal, but never equal.

You can probably see the "simmetry" too. I have detected three types.

I had to stop at this stage, as 16 bits of binary representation wouldn't let me go further. My next challenge was to see if the pattern remained with 32 bits. I was quite confident it would also show up with that number of bits.

And as you can see... did! The pattern goes on with the 14.112 first prime numbers. And it looks as if it will continue working further.

I had to stop here due to image resolutioin issues. I'll try to work on a higher resolution one later. By the way, I tested it with 64 bits, and the pattern remains.

Here you can see a high resolution version (1980x1024), that showas primes up to 799.999.

The pattern remains, and we can now make some considerarions about it:


I first put this up in my web on august 18th of 2011 at 2:00 am, in Loeches, Madrid, Spain. I'm not sure of the implications the discovery of this pattern might have. In my basic mathematical knowledge I think no on has found a visual pattern in prime numbers before, but I could be wrong. Nevertheless, after some months of research I'm thrilled to have found one!