There ARE a number of messages that can be represented as a polynomial of prime powers, look up Godelization. More generally there are SOME very compact methods

There ARE a number of messages that can be represented as a polynomial of prime powers, look up Godelization. More generally there are SOME very compact methods to write large numbers using functions, eg factorial and tetration: 12345! + 3^^3^^2 -1 would be huge. The problem is MOST numbers do not have compact representations so they'd be bigger. I do wonder about how effective it is to pad messages in-band to align with the nearest compact representation. Funny you should mention scifi because there's a story that got me interested in this topic [1] where some folks encode a large, important message into a compact algebraic expression like this. Turns out they're pissed at the recipients and the decimal expansion would be enormous so it would take them a while to decode it. 1. Fredrick Pohl, "Starburst", pp 56, 58, 59.