An Interesting Pattern in the Prime Numbers: Parallax Compression

(UPDATE: Thanks to comments from readers we have found that the pattern does not exactly match the GCD triangle for some values of the number of cells and rows: this possibly makes it a more interesting finding. Join the discussion in the Telegram group as well – details below) .

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. Here is a more robust Github version of the code in a Mathematica notebook if you want to explore it yourself (note: Thanks to Stephen Wolfram for taking a look at the Mathematica code and advising us in January, when we were wondering whether this might break crypto and needed advice; the answer is no, it doesn’t break crypto, but Mathematica is pretty great!).

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, the width of a cell, n, is 100, so each cell represents 100 integers in the sequence, and the pattern holds for 100 rows.

Initially we found that this pattern matches a known numerical sequence,OEIS A054521 — and it recurs for other even values of n, so it is self-similar at various levels of scale.

For example, If n = 50, then each cell represents 50 integers, and the pattern holds for 50 rows. If n = 200 then each cell represents 200 integers, and pattern holds for 200 rows.

Here is an animation (Thanks to Ian Rust) that shows the pattern approaching the GCD sequence pattern, as the values of n increase.

However, some readers noted today in the thread on Hacker News that GCD doesn’t hold for all values of n.

For example, for odd values of n we see a different pattern that is also rather interesting. Here is n = 99:

Like the GCD pattern we see for even values of n, the odd valued n pattern also recurs for different sized odd values of n. This means that this pattern is not simply the GCD sequence — there are variances that we don’t understand yet.

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 and eventually we made a little progress finding at least one known sequence that generated the pattern for even values of n, without needing any primality testing. But as noted above, it doesn’t hold for all values of n, and we have not done a formal proof nor have we tested a large set of values of n and compared results.

We’re not exactly sure what this all means yet — it might not mean much — it might just be a pretty visualization — 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, in case there is something more to this.

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? The self-similarity at various levels of scale, and the fact that it isn’t fully described for all values of n by a known sequence means there may still be more to understand about this.

In general, finding any kind of non-random pattern in the distribution of primes is potentially interesting. 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 are not mathematicians, but hopefully some mathematicians reading this will take it further than we can.

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).

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Nova Spivack

Venture Producer