An Interesting Pattern in the Prime Numbers: Parallax Compression

The drawing is incorrect.<p>More specifically, it’s a drawing of OEIS A054521 (black if gcd(row, col) == 1, red otherwise), not of the parallax compressed primes. The two drawings do not match as claimed. The first place where they differ is row 9, column 1, which is drawn as black even though none of 217, 226, 235, 244, 253, 262 are prime.<p>It’s clear that gcd(row, col) == 1 is a necessary condition for there to be any primes in that cell (which consists of the numbers col + 3row² − 3row, col + 3row² − 2row, col + 3row² − row, col + 3row², col + 3row² + row, col + 3row² + 2row), but it’s not sufficient. There’s no way it could be sufficient, because there are only constantly many (six) numbers tested in every cell, but the asymptotic density of the primes goes to 0 and the asymptotic density of A054521 goes to 6&#x2F;π².

I wonder why the authors took the time to generate an image of their pattern but refrained from giving the actual DEFINITION. This does not foster constructive discussion of any possible ideas present. As can be seen in dozens of well-meaning comments here, people waste time reversing and guestimating parameters etc.<p>Looking at the layman letters that my institute gets on a regular basis, I can say, that this is unfortunately a recurring theme with amateur mathematicians: They fail to state their basic definitions and assumptions and seem all to eager to dive right into applications, be it computer graphics, cryptography or finance.<p>More to the point: This picture seems from a cursory inspection to plot T(k,n) with n=row, k=column from the top left. But why is this interesting?

Awesome! It&#x27;s the math enthusiast&#x27;s dream to come up with something new and exciting outside of academia. Recently I discovered what I thought was an interesting chaotic map, but after posting a question about it to Math StackExchange[1] and emailing one or two professors (no response, which is understandable), my obsession waned and I gave up on trying to figure out if it had any significance. Maybe I should keep trying!<p>[1] <a href="https:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;2654984&#x2F;identifying-this-chaotic-recurrence-relation" rel="nofollow">https:&#x2F;&#x2F;math.stackexchange.com&#x2F;questions&#x2F;2654984&#x2F;identifying...</a>

Yep - we are not claiming to be mathematicians, we&#x27;re pattern hunters... this is an open invitation to others with more expertise to chime in and help figure this out... it&#x27;s possible it is a minor discovery or even not a discovery... or it could be useful or even very useful. We don&#x27;t know. Please help us explore it!

This made me curious, so I wrote a javascript version that renders a larger image of it:<p><a href="http:&#x2F;&#x2F;www.gibney.de&#x2F;parallax_primes" rel="nofollow">http:&#x2F;&#x2F;www.gibney.de&#x2F;parallax_primes</a>

The way the numbers are picked into the 75-number groups is responsible for pretty symmetry.<p>If you replace condition isPrime() with simpler checks: &quot;is not divisible by 2&quot; &quot;is not divisible by 2 and 3&quot; .. &quot;... by 2, 3 and 5&quot; .... &quot;... by 2,3,5,7,13,17 and 23&quot; ... you&#x27;ll get more and more complex images but still symmetrical.<p>For me the whole thing is subtle hiding of messiness of primes into the strong, pretty, symmetrical shape which obscures the mess and just gets richer and more artistic due to that.<p>Experiment with the code provided by user no_gravity here: <a href="http:&#x2F;&#x2F;www.gibney.de&#x2F;parallax_primes" rel="nofollow">http:&#x2F;&#x2F;www.gibney.de&#x2F;parallax_primes</a><p>By changing the contents of isPrime() function you can see how you get fooled into thinking there&#x27;s order in primes by mixing messiness of primes into the order of number picking scheme.

My take, but correct me if I&#x27;m wrong.<p>Take the relation to the GCD triangle <a href="http:&#x2F;&#x2F;oeis.org&#x2F;A054521" rel="nofollow">http:&#x2F;&#x2F;oeis.org&#x2F;A054521</a><p>At GCD(n,k)!=1 all numbers are divisible by GCD(n,k) therefore contains no prime<p>At GCD(n,k)==1 we have <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dirichlet%27s_theorem_on_arithmetic_progressions" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dirichlet%27s_theorem_on_arith...</a> - so those series contain infinitely many primes - and seems like they actually contain at least one prime in all the pixels of the first N rows (but this should be explained&#x2F;proved, if it is always true for any chosen N, or just happen to be true for the N-s tried by the OP)

It wasn&#x27;t exactly clear to me when I read the article, so here&#x27;s my explanation:<p>- take a line of integers, color them black if prime, red otherwise<p>- hexagonally arrange them in a spiral (similar to Ulam&#x27;s spiral)<p>- cut the hexagon into six equilateral triangles<p>- overlap the triangles (rotate where necessary)<p>- if any pixels are black, color the whole thing black, if none are black, color it red<p>- interesting pattern arises<p>- interesting pattern already exists as per <a href="http:&#x2F;&#x2F;oeis.org&#x2F;A054521" rel="nofollow">http:&#x2F;&#x2F;oeis.org&#x2F;A054521</a><p>Edit: They may be packing more than 6 numbers into a pixel, looks like 75. Unsure how that would look visually, but you extend the above to use 75 or any other number. Unsure why 75 was chosen, maybe it&#x27;s the only interesting one?

I wrote some code to replicate this in the console, overlaying A054521 (in colors) onto their triangle (using _ and #) for any N. The output from running<p><pre><code> .&#x2F;prime-triangle.py 74 </code></pre> is included as x_output_74.png. It&#x27;s not fancy, but it could save you some time trying to figure out what the actual formulae are.<p><a href="https:&#x2F;&#x2F;gist.github.com&#x2F;mortehu&#x2F;ccca0bafc7a9caa26d6008379057e3cc" rel="nofollow">https:&#x2F;&#x2F;gist.github.com&#x2F;mortehu&#x2F;ccca0bafc7a9caa26d6008379057...</a><p>Edit: Other than 2, 14, 20, 30, 38, 44, 50, and 74, the two patterns match perfectly for every even N up to at least 600.<p>Edit 2: Looks like this is related to Linnik&#x27;s theorem: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Linnik%27s_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Linnik%27s_theorem</a>

From a cryptography standpoint, could this hint at attack vectors for things like discrete log problems? I&#x27;ve only learned of the math behind that myself recently, not sure what implications having a &quot;topographical map of the primes&quot; could have, especially if the pattern is relf-repeating regardless of the size of the primes.

My gut sense is that this has something to do with visualizing the sieve of Eratosthenes, and nothing more than that. But I&#x27;d be happy to hear otherwise.

A few quick observations:<p>- The right edge of the triangle is always red (ie, no primes present), because it represents (row number) * (3* (row number)+[1..6])<p>- Prime numbered rows are always black (except the far right column). I can sort of feel why this is true but can&#x27;t express it mathematically yet.

&gt; on January 18, 2018, I found a numerical sequence that generated the exact same pattern as Shaun’s pattern<p>Does this mean that we have a sort of bloom filter-esque test for primality? (ie, it will give you a guaranteed no in O(1) but you&#x27;ll have to crunch numbers to get the yes?)<p>If so, are there implications for things that want to know &quot;is it prime?&quot; quickly? Crpytography comes to mind, for instance...

Shaun is going to update the code with some additions that will hopefully help clear up some of the questions &#x2F; confusion below. We will also post Mathematica code that is more rigorous. And a new finding. Stay tuned.

What would it look like if instead of a binary coloring, you used a gradient coloring representing the number of primes in each range?

This just shows the reduced residues mod n. There are n squares in the nth row from the top. Color the kth square of the nth row black if gcd(k,n)=1, color it red otherwise.

Just run a search on: prime numbers fractal, and many interesting articles (both academic and other) will surface. Search for: zeta function fractal, and you&#x27;ll find more. I&#x27;m no mathematician but it wouldn&#x27;t surprise me if two very deep fields turn out to be different &quot;manifestations&quot; of some deeper underlying truth, and I then wonder about the possibility of a larger grand unified theory that includes both maths and physics that would have far reaching insights which would go beyond both mathematics and physics, but then I realise that perhaps I&#x27;ve seen too many movies :-)

These runes would serve excellently as a written alphabet for an alien species in fiction (or maybe for modrons or other Lawful creatures in D&amp;D).

The Javascript code at Observable has been updated to render it correctly for both even and odd values of n now.<p>This addresses the issues that were pointed out below for the most part.<p>In short, actually there are some curious patterns in this, and they are not simply equivalent to OEIS A054521 (as we, and others, initially thought they were).<p>For even values of n, they can be rendered by the GCD sequence, without the primality testing. But for odd values the pattern is different and GCD doesn&#x27;t describe it.<p>The Mathematica code on Github lets you experiment with this.<p>There are some nice versions, and experiments in different ways of coloring or arranging the graph in the Telegram group as well.<p>Telegram group <a href="https:&#x2F;&#x2F;t.me&#x2F;joinchat&#x2F;G8AnchIna2q8yn1lGHirkA" rel="nofollow">https:&#x2F;&#x2F;t.me&#x2F;joinchat&#x2F;G8AnchIna2q8yn1lGHirkA</a><p>Javascript code <a href="https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-ulam-spiral" rel="nofollow">https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-u...</a><p>Mathematica code: <a href="https:&#x2F;&#x2F;github.com&#x2F;shaunxcode&#x2F;a-pattern-in-the-primes" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;shaunxcode&#x2F;a-pattern-in-the-primes</a>

Is there are particular reasoning or meaning to each dot being 6 numbers? Is there any significant changes if you pick other numbers per dot, following the same pattern?<p>Based on the linked explanation here: <a href="https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-ulam-spiral" rel="nofollow">https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-u...</a>

UPDATE - Saturday May 19<p>Join the Telegram group to discuss: <a href="https:&#x2F;&#x2F;t.me&#x2F;joinchat&#x2F;G8AnchIna2q8yn1lGHirkA" rel="nofollow">https:&#x2F;&#x2F;t.me&#x2F;joinchat&#x2F;G8AnchIna2q8yn1lGHirkA</a><p>Note that Even and Odd Values of N have a very different pattern. For example try using the values 99 and 99, and then 100 and 100, in this HTML preview version:<p><a href="https:&#x2F;&#x2F;htmlpreview.github.io&#x2F;?https:&#x2F;&#x2F;github.com&#x2F;acmegeek&#x2F;primes&#x2F;blob&#x2F;master&#x2F;index.html?11" rel="nofollow">https:&#x2F;&#x2F;htmlpreview.github.io&#x2F;?https:&#x2F;&#x2F;github.com&#x2F;acmegeek&#x2F;p...</a><p>Here is animation of increasing even values of N approaching GCD: <a href="https:&#x2F;&#x2F;streamable.com&#x2F;l7r96" rel="nofollow">https:&#x2F;&#x2F;streamable.com&#x2F;l7r96</a><p>CODE TO TRY:<p>Javascript <a href="https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-u.." rel="nofollow">https:&#x2F;&#x2F;beta.observablehq.com&#x2F;@montyxcantsin&#x2F;unwinding-the-u...</a>.<p>Mathematica <a href="https:&#x2F;&#x2F;github.com&#x2F;shaunxcode&#x2F;a-pattern-in-the-primes" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;shaunxcode&#x2F;a-pattern-in-the-primes</a><p>Perl <a href="https:&#x2F;&#x2F;www.dropbox.com&#x2F;s&#x2F;z5tfub5geyuctex&#x2F;prime-draw.pl?dl=0" rel="nofollow">https:&#x2F;&#x2F;www.dropbox.com&#x2F;s&#x2F;z5tfub5geyuctex&#x2F;prime-draw.pl?dl=0</a><p>EXPLANATION:<p>In short, actually there are some curious patterns in this, and they are not simply equivalent to OEIS A054521 (as we, and others, initially thought they were).<p>For even values of n, they can be rendered by the GCD sequence, without the primality testing. But for odd values the pattern is different and GCD doesn&#x27;t describe it.

Just a heads up, but I would pay money for mugs&#x2F;t-shirts with the pattern. Of course I can just use to code to make it myself, but it&#x27;s nice to support people sometimes

We will be happy with &quot;it&#x27;s a really pretty visualization of the primes&quot; or &quot;it&#x27;s an improvement on the Ulam Spiral.&quot; But if it has more value than that (not sure.. but possibly there is some link in this that might be useful) then that&#x27;s great too.

I am a little confused. When it says that each square corresponds to n sequential numbers, does it mean that the first square is the first n, and the second square (which is on the second row) is the first n after those, and the third square (also in the second row) has the first n after those, and so on,<p>Or, do the regions overlap?<p>I tried to look at the first 4 rows for n=5, but did not see the pattern depicted (each interval had a prime in it).<p>Am I interpreting what is meant by the blocks incorrectly, or does the pattern not work for small enough n?

The Ulam Spiral: <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Ulam_spiral" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Ulam_spiral</a>

Here is a Dropbox link to the Mathematica code. Thanks to some of the input today we have found that this pattern does not exactly match the gcd triangle - which actually makes this more interesting in a way.<p><a href="https:&#x2F;&#x2F;www.dropbox.com&#x2F;s&#x2F;d2dfwhxdmzkp4y4&#x2F;a-pattern-in-the-primes.nb?dl=0" rel="nofollow">https:&#x2F;&#x2F;www.dropbox.com&#x2F;s&#x2F;d2dfwhxdmzkp4y4&#x2F;a-pattern-in-the-p...</a>

The Telegram link leads to a distribution channel, not a group, so one cannot actually discuss it there. I let the author know through the contact feature on the website, as it might be unintentional.<p>Edit: The link was changed to a group (at the bottom of the post).

Maybe it is good to know that within the sequence of all prime numbers there infinitely wide gaps : take (n+1)! +2, (n+1)! +3, ..., (n+1)! + (n+1)<p>That is a sequence of n consecutive numbers none of which are prime.<p>Not sure how that would map in this triangle shape yet.

For those that like to mess around with the J programming language, here is what I believe is correct code to generate this parallax compression pattern.<p><pre><code> pack =: 3 : &#x27;+.&#x2F; (r,y+1) $ (npc*y+1) {. (npc*y) }. p&#x27; &quot;0 pc =: 3 : 0 r =: npc =: y c =: +&#x2F; 1+i.y p =: 1 p: (1+i.c*npc) pack i.r ) </code></pre> Image of output for numbers-per-cell = 75 here: <a href="https:&#x2F;&#x2F;twitter.com&#x2F;seanstickle&#x2F;status&#x2F;997675789264015361" rel="nofollow">https:&#x2F;&#x2F;twitter.com&#x2F;seanstickle&#x2F;status&#x2F;997675789264015361</a>

This is awesome. How can we access the code to play around with?

I&#x27;m noticing a few neat things about there is left-right reflected symmetry.<p>Really neat! (I still know I haven&#x27;t the math chops though. Good place to further my study)

Here is an animation (thanks to Ian Rust) showing how the even values of n approach the GCD pattern (OEIS A054521) as the values increase <a href="https:&#x2F;&#x2F;streamable.com&#x2F;l7r96" rel="nofollow">https:&#x2F;&#x2F;streamable.com&#x2F;l7r96</a> however the odd values of n are not equivalent to the GCD pattern.

Is each row the same as if the remainder on division of the cell number by the row number is 0 then red, otherwise black?

Is this really capturing a pattern in primes, or a pattern in non-primes (which is totally not surprising).

“Fooled by randomness.”

The prime numbers are interesting no matter how you slice and dice them!

I wonder if there&#x27;s a pattern to how many primes per cells-with-primes.

It may turn out to be a pretty small part, but history was made today :) keep it up, stuff like this is awesome! You may not be mathematicians and it may turn out this is nothing hugely new, but even so, the proof of such itself is interesting. keep up the cool work!

This is... Amazing...!

in an Hexagon is a pattern following rules of the fibonacci spiral... but never told anybody about it cause of the effect....

Does this get us closer to being able to generate the Nth prime without factorization &#x2F; primality test of every number?

If you throw out the last cell of every row, the pattern is symmetrical.

&quot;It looks like something they found on the ship at Roswell.&quot;

The pattern looks nice, but I would have asked some mathematicians before announcing this as something novel.

If this can really map prime numbers to a least a general region, would we be able to break Diffie-hellman key exchange more quickly?<p>If so this could be a huge blow to security.<p>But I must say this is amazing that they were able to visualize prime numbers in this way. These guys are geniuses.