The digital sum of a number is the sum of its digits, and it's similarly defined for digital products.

Let's define the digital exponentiation of a number to be the evaluation of the exponentiation of all of the digits written in a tower. So for example:

$f(345) = {3^4}^5 = (3^4)^5=81^5=3486784401$

For some values, like 200, this will obviously converge to a single-digit number, and the digital exponentiation of a single digit is just that digit. For others, it appears that they will diverge infinitely.

However, is there a way to prove that it will always either converge to a single digit or diverge infinitely? As in, does it ever stabilize at some other value(s), or form a cycle?


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