Illegal numbers are those numbers that contain information, i.e. encode it, which is illegal to disseminate or possess. These are not numbers that are truly prohibited by law as numbers, but numbers that can be used as containers of information that the law prohibits from possessing or disseminating. Let’s see in what sense a number can contain information, how the first illegal prime number was born, and how the number 2117565345758066597926955075797379264 which gave rise to the flag symbol of freedom of speech.
What are illegal numbers and their history
In what sense can a number contain information? It is thanks to cryptography, that is, replacing a word with other symbols – numbers in this case – that represent it.
Let’s take an example, suppose we want to communicate the word “hello” using only numbers, a rather simple way of doing this is to replace each letter with the number that represents its position in the alphabet: c=3, i=9, a=1, o=15, so hi = 39115 (according to the international alphabet with 26 letters). The number we created, 39115, can be converted back to the word “hello” by anyone who knows the conversion method used.
Numbers, in particular binary numbers, are the language used by computers, and computer systems in general, to manage information and the word digitization refers precisely to the conversion of information of all kinds – for example texts, sounds, images… – into binary numbers.
Since any type of information can be digitized, then any illegal information – to be possessed or disseminated – can also be digitized giving rise to potential illegal numbers.
But have some numbers really risked becoming illegal? In some ways yes, in the IT field, following the court case concerning Jon Johansen, a Norwegian teenager who in 1999, together with two anonymous people, created and made available on the Internet a program called DeCSS which served to illegally unblock the viewing of commercial DVDs.
When the program was shared on the Internet, the Motion Picture Association of America reported Joahnsen to the Norwegian police and the boy was put on trial. At the same time as the judicial process experienced by Johansen, protest episodes aimed at criticizing the freedom to disseminate computer program codes multiplied.
The DeCSS code was then converted and disseminated in the most bizarre forms, including printed t-shirts and poems, many of which were collected in a gallery published, and still visible, by the American academic David S. Touretzky of the Computer Science Department of Carnegie Mellon University. As part of this spontaneous protest movement, the idea of representing the DeCSS code, among other things, also in the form of a number was born.
Carmody’s prime number
As part of the aforementioned protest movement, Phil Carmody, an engineer from Cambridge, intervened with a somewhat winning idea:
if we could codify DeCSS by a number that is significant to the advancement of mathematical research then this number and its dissemination could not be prohibited by any law.
Carmody then decided to look for a way to represent DeCSS with a prime number that was relevant from a mathematical point of view and succeeded by finding what is considered the first illegal prime number in history, a number of 1401 digits.
This number, at the time it was found by Carmody, was among the largest prime numbers ever found using a particular formula and, according to Carmody, the number had therefore earned itself a small place in the history of the advancement of mathematics, enough to justify sharing it via the Internet, protecting it from accusations of illegality.
But Carmody’s prime is not the only example of an illegal number. A few years later, in 2007, the Advanced Access Content System (AACS) License Administrator and the Motion Picture Association (MPA) took legal action against anyone who published the number 21175653457580665977926955075797379264 which was used as a key to encrypt Blue-Ray and DVD discs.
Again the idea that someone could own a number and forbid its diffusion resulted in the opposite effect and this number spread more and more and in various forms, such as song lyrics, poems and T-shirt prints. Indeed, converting this number to base sixteen we obtain the hexadecimal number 09F911029D74E35BD84156C5635688C0 which was used to design the flag symbolizing freedom of speech that we show in the image below.
