Illegal Numbers (and where to find them)

This article was written by one of my students, Matthew Stephenson. If anybody else would like to contribute anything then get in touch. It doesn't have to be mathematical in the slightest; it only has to be interesting to me and I'm a pretty curious person. Enjoy.

Illegal numbers, as the name would suggest, are numbers where the possession or transmission of could in theory result in prosecution. These numbers can be expressed in standard, decimal, notation, as well as by other means.

The idea of illegal numbers is based on the idea that all information can be expressed as a number. While in the real world such numbers representing things have to either be extremely large or the objects themselves would have to be simplified in order for the number to be manageable (think mapping the human genome), in the digital world these numbers can often be expressed in very few digits. So, if you understand that information exists which it is illegal to possess or distribute, there must be numbers that exist that it is similarly illegal to know, view, etc.

Examples come up in cases such as the key for (partly) “cracking” DVD and Blu-Ray players, which can be expressed as 38 digit number (13,256,278,887,989,457,651,018,865,901,401,704,640).

This isn’t the only way numbers such as this can be expressed, however. Converting to hexadecimal (base-16) gives 09F911 029D74 E35BD8 4156C5 635688 C0. This is all one number, but I’ve grouped it into chunks of 6 digits for ease of the next step. Numbers in hexadecimal can similarly easily be converted to colours; 6 digits make up each colour, two each for the red, green and blue components of it (RGB). Thus exactly the same information as that 38 digit number can be transmitted in 5 colours and two additional digits, with no data being lost in the process, making anything displaying those five colours and two characters illegal to own, in theory. You can obtain shirts and flags with this pattern on, by the way, and the design is shown below.

As a side note, the only reason that this is known now is because the owners of the system and the key for cracking it tried to remove it from the internet. The Streisand effect soon followed, the principle that if you try and remove content from people, it will become more and more widespread, far more than it was before it was originally attempted to be censored, reversing the original intention of the people trying to remove the data.

Keys such as this are not the only things can be expressed like this, however. Everything stored on a computer is stored as a (albeit usually long) binary number. This makes the number of proprietary software just as illegal to own without permission. Similarly, images and videos are also stored as numbers inside a computer. And given the limitless different ways of making symbols, colours, etc. represent different numbers, you could argue possessing any such symbol or colour a violation of intellectual property, though any such court ruling would, in most likelihood, cause protest.

I find myself using these ideas in order to point out the difficulty of the laws surrounding intellectual property, and what it actually means to own something in a system where exactly replicated data is commonplace.

However you like to look at it, the fact remains that this sentence, all of the text inside this article, and indeed this webpage, including all of its pictures and data on fonts, etc., can simply be expressed as a number.