A number that represents some information which is illegal to posses or transmit, making said number technically illegal. It can also refer to numbers that have a particular meaning or connotation that a government wishes to censor.
When focusing on some specific class of numbers one could create funny illegal numbers such as:
- Illegal primes
- Illegal Pythagorean triples
- Illegal triangular numbers
- Illegal Fibonacci numbers
- etc. This could be declined for any infinite sequence of numbers in which you could encode such illegal information.
Interestingly, there has to be some criterion on whether this information is easily recovered from its encoded form. Otherwise any normal number would qualify as illegal, maybe even \(\pi\).
In practice this notion of making numbers illegal makes little sense. For example, one could always evade countermeasures by changing a few bits in the underlying data or source code and use that new number. One can also use various compression, or simple encryption methods (rot13) to change the data enough that it can be easily recovered but represents a completely different number. Most attempts at censoring numerical data have hardly succeeded because it is relatively easy to change their representation.
Of course because of the nature of numerical data, illegal numbers can be encoded into images, videos, text, or any data format making it very hard to actually enforce their illegality.