Early this year a software engineer, Shaun Gilchrist, reached out to me after reading a blog post of mine from many years ago, about my informal search for hidden patterns in the prime numbers.

The Ulam Spiral revealed non-random patterns, but they didn’t quite match up. Both Shaun and I had long felt there was a better way to wrap the primes that would reveal a deeper structure.

Shaun explained that he had developed a new algorithm (he calls it “Parallax Compression”) for wrapping the primes on a plane, and visualizing their distribution, inspired by the Ulam Spiral.

After his initial discovery, Shaun searched the Web for anyone else who was thinking this way and that led him to my blog post, and to me.

Shaun’s algorithm reveals an interesting non-random, fractal-like pattern in the distribution of primes, that to our knowledge, has never been seen before.

It makes it possible to easily see where there are regions of prime and non-prime numbers, anywhere on the number line, at any level of scale.

When one looks at a visualization of this pattern, it appears reminiscent of runes, Mayan glyphs, tapestries, and hieroglyphics. If you look at it for a moment or two you will see there are several levels of nested geometric shapes within it that appear to have a kind of fractal symmetry:

 

A cell is colored black if there is at least 1 prime number within it, and red if there are no primes within it.

Here, n = 75, so each cell represents 75 integers in the sequence, and the pattern holds for 75 rows.

Interestingly, this same pattern holds for more numbers, and for different intervals. It does not repeat as you add rows, yet continues to be self-similar.

For example, If n = 1000, then each cell represents 1000 integers, and the pattern holds for 1000 rows. If n = 100,000 then each cell represents 100,000 integers, and pattern holds for 100,000 rows.

This algorithm also reveals sequences (that we call “runs”) of primes and non-primes along various axes that might be useful for predicting prime and non-prime regions.

After Shaun reached out to me with his discovery, we spent many sleepless days and nights collaborating to see if there were even deeper patterns behind this new visualization.

After hammering on it with everything we had, on January 18, 2018, I found a numerical sequence that generated the exact same pattern as Shaun’s pattern, without all of the primality testing.

We’re not exactly sure what this all means yet — but it’s interesting enough (to us at least) that we decided eventually to make this public so that others could help us explore it further.

Perhaps this is a topographical map of the distribution of the prime numbers? Perhaps this might be useful in number theory, or in some area of science?

Are there connections between this and other research findings, such as this recent article we found on aperiodic order in the primes?

We don’t know yet, but we are curious to find out.

We hope you enjoy this, and if you make further progress on this, or find anything that may be connected, please let us know. (You can discuss it with us, and others who are interested, on this Telegram group).