bookmarking to think about later... does this hold for representing numbers as one base compared to another?
Regarding a prime as having higher entropy / less structure than say a perfect square or highly divisible number
a prime is a prime in any base, but the number of divisors will differ in non-primes, if the number is divisible by the base then it may appear to have more structure (smaller function necessary to derive, kolmogorov style),
does prime factorization have anything to do with this? i can almost imagine choosing a large non-prime whose divisibity is only obvious with a particular base such that the base becomes the secret key - base of a number is basically specifying your dictionary, no?
I don't think there's any interesting difference with different bases. Usually the base you represent stuff in is a relatively small number (because using very large bases is already wildly inefficient). I think it only usually makes sense to consider constant or logarithmic bases. If your base is scaling linearly with your number then things are going to be weird.
The problem of finding factors is only complex when you're asking about relatively big factors. If you're looking for constant or log sized factors you can just do trial division and find them.
Changing the base of number representation with a random basis feels like XORing a string with a random string, which is to say you're adding entropy equal to the random string. My thinking is that for any number representation M, you can get any other number representation N given a well-chosen base. So when presented with the encoded N, the original number could be any other number with the same number of digits. But once you put reasonable bounds on the base, you lose that flexibility and end up adding negligible entropy.
Regarding a prime as having higher entropy / less structure than say a perfect square or highly divisible number
a prime is a prime in any base, but the number of divisors will differ in non-primes, if the number is divisible by the base then it may appear to have more structure (smaller function necessary to derive, kolmogorov style),
does prime factorization have anything to do with this? i can almost imagine choosing a large non-prime whose divisibity is only obvious with a particular base such that the base becomes the secret key - base of a number is basically specifying your dictionary, no?