Say I have a positive integer that is one-thousand digits long. What math could I use to represent this number as a series of exponents in a significantly shorter form than the original number? The number is rather random.
For example, $(4^5)^6$ makes 1152921504606846976. What steps can I use to deduce that 1152921504606846976 can be composed of $(4^5)^6$?
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Given $x_0$ you are trying to find $x_1$ and $y_1$ such that $x_0={y_1}^{x_1}$. And then you want to repeat the process with $x_1$. Most often this process has only the trivial solution of $y_1=x_0$ and $x_1=1$. Otherwise you can attempt the following:First: Find the prime factorization of $x_0={P_1}^{\alpha_1}\cdots{P_n}^{\alpha_n}$. Second Step: Find GCD of exponents $\alpha_1 \cdots \alpha_n$. This will be your $x_1$. Now you repeat the process.
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If it is hard to find the factorization of the number into primes, then you can instead test whether it is a perfect square, cube, 5th power, 7th power, etc. There are efficient ways to do that --- a websearch for "integer square root" will get you started. If your number is $n$, and it turns out to be a perfect square, say, $n=m^2$, then you look at $m$, testing to see whether it is a square cube, etc.
For your particular number, this approach would find $n=m^2$, $m=p^2$, $p=q^3$, $q=2^5$, from which you can put together your short representation.
Fundamental Theorem of Arithmetic
Your number factors to $2^{60}$
You can use the factorization however you'd like to create a smaller number of numbers by multiplying factors together.
Regards