Wondering what the way of producing the shortest length string form of the natural numbers $\mathbb{N}$. I am thinking about this in terms of a binary representation (base 2 representation), but I think I could figure that out starting from a standard base 10 representation. It seems like this has something to do with polynomial equations, so I started with that. (Not totally sure I'm doing it right)....
$1 = 0^1$
$2 = 2^1$
$3 = 3^1$
$4 = 2^2 = 4^1$
$10000 = 10^4 = 100^2 = 10000^1$
$1000000 = 10^6$
At larger numbers it becomes apparent that you can represent them with smaller "strings" than the original number. So 1000000 is 7 digits, but $10^6$ is only 3. If the number was large like $10^{50}$ then it would be $10000000000000000000000000000000000000000000000000000...$, so there is a big gain.
Some other thoughts...
For even larger numbers you can perhaps start nesting the exponents, so $10^{50^{50}}$, and gain an even larger advantage.
Basically I would like to know if there are any equations or systems for figuring out the ideal "smallest" representation of the number as a polynomial equation, where the number could be represented as any base. So $1000000$ is better represented as the more compact $10^6$ since it's only 3 digits instead of 7. But I don't see any equations here on how to figure this out (yet).