Primify

Starting search for prime.
Note that in this instance we would expect to run
about 13799 primality tests.
Found!
Did ~2291 primality tests
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Primify takes an image and embeds it into a prime number, so that, when viewed from a distance, it resembles the original image. It was created by Levi Borodenko, a masters student in mathematics and statistics at the University of Oxford. While it works particularly well for logos and simple line art, was actually first developed for portraits.

The idea came while I was a counselor at PROMYS -- a summer camp providing an immersion into rigorous mathematics. We had a secret Santa and me being a poor student, I tried to come up with a gift idea that would light a maths majors heart without being pricey. So I decided to find a prime that looked like the face of the person that I had to gift. I coded up a quick and dirty script (back then I was using python to find the number and then mathematica to find the next prime). After doing this for a few years I decided that more people should be able to enjoy generating these primes. This is when I coded up the open-source tool.

In primify, the underlying data is not only used in a graphical way, but also holds information when read as text; in this case, a single prime number. This is multicoding, where content holds multiple meanings when read in different systems. It's a strategy familiar not only from esolangs like legit, which multicodes between git commands and code, and polyglots, where one program executes successfully in multiple programming languages. The strategy here, where the text functions as a picture, is more akin to how multicoding functions in obfuscated code that builds on concrete poetics, where the opacity of the code is offset with a visual clue as to its function. We can see this in the example cited here by Adrian Cables, or the cauldron example in this post about the IOCCC. The term multicoding comes from the seminal paper on esolangs and code art, "A Box Darkly" by Matteas and Montfort (as "multiple coding").

Where obfuscated code can use whitespace (quite indiscriminately when it's in C), this is not an option when the text is a number. Primify has access only to the ten digit symbols, which are presented without breaks -- so its visual strategy comes from ASCII art, where each pixel (after resizing) is represented by one character. The choices of digits selected by their relative density, to represent different shadings. Borodenko uses only the digits with the most distinct grayness values: 1, 7, 3, 9, and 8 (in that order). To prepare the image, primify runs image processing steps like edge enhancement and smoothing, and then quantizes the image into grayness levels corresponding to the palette of those five numbers.

But, as Borodenko points out, not only does ASCII art have a large variety of different characters ("," and "@" instead of just "1" and "7", etc) but also usually turns images into strings of the order of 25,000 - 40,000 characters. This would create far too large a number to find a close prime in a reasonable amount of time, and so creates an additional challenge for primify.

For primify the numbers should not exceed 17,000 as otherwise it is simply very hard to find a nearby prime as the primality tests take a long time and more of them are needed. Thus, we face the problem of creating a small resolution ASCII art using only numbers. Here is where only using 5 characters comes in. 5 and 6 for example create a very similar shade and with the small space that we have, that simply does not provide a visible difference.

I use sympy for primality checking. If the number is larger than 2^64, then a strong BPSW test is performed. While this is a probable prime test and we believe counterexamples exist, there are no known counterexamples. So if primify returns a prime that is actually not a prime, you get to publish a paper as a consolation price ;)

The images with simple line art seem to work the best. The strategy for how the edges are determined can be set with the "method" argument, but images with large fields of color (I attempted several Josef Albers Homage to the Square paintings) seem to gain excessive edges due to the enhancements. Logos like the github one above work quite well, as do some portraits.

The size of the image determines how long it takes to find an appropriate prime. If you want a very large prime, to get the most density of digits, you may be in for a long wait.

It may very well be that the next number after the generated number is prime so the tool may run VERY quickly. However, it is known that the gaps between primes are unbounded. Take for example n = 2 x 3 x 4 x .... x 1000 + 1, then the next 1000 number will not be prime as n+1 will be divisible by 2, n+2 by 3 etc.. So the program may take a loooong time. Thankfully we know that there are infinity many primes - so at least we have a guarantee that the code will halt :P

In my experience, the numbers have about 5000 digits to be reasonable easy to compute in <30min. between 7000 and 10000 may take 1-3h or even longer if you are unlucky.

Apart from primify, Borodenko developed Best Circle, a fun app that evaluates how well you can awkwardly draw a circle with a mouse (for people who don't use drawing tablets). And has a great landing page for his website; net artists take note!

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Jonas Mekas in clown hat and clown nose (squint and you can see it!), primified