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A fractal crystal lattice of Gaussian primes, or a result of interpolation?

Posted 2 months ago
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While researching the possible fractality of prime numbers, I happened upon what I have been calling a fractal crystal lattice of Gaussian primes. Credit to Dr. Eric Weisstein for the concept and the Mathematica code, which is from the MathWorld website. I ran Dr. Weisstein's notebook with truly absurd numbers of Gaussian primes, some of which took hours to compute, and produced a series of images with an obvious crystal lattice-type pattern. I provide raw screenshots and the edited notebook on my website: https://www.andrewwatters.com/primes/images/

I am wondering whether this is a real pattern or a result of artifacts in the image processing pipeline in Mathematica. It seems doubtful that it's in the image processing pipeline given the clear recurrence of various similar patterns at varying image sizes. I gave up at r = 10,000 which caused Mathematica to freeze when trying to save the plot as a separate file. The pattern may not be obvious in the following image due to compression, but if you check out the raw images, it's definitely there. enter image description here

Might want to check section 9.2 of Computational Number Theory by David Bressoud and Stan Wagon. It has the picture in Out[86] of the notebook you link to, and discusses some aspects of Gaussian primes.

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