Which is the largest power of natural number that can be evaluated by computers?

Question

Which is the largest power of natural number that can be evaluated by computers? For example if we take a very large power of 7: $7^{120000000000}$. Can a computer calculate this number?

Answer 1

That number will not be a problem. Using elementary methods, we estimate that it will have around 101 billion digits. It's easy to store 2 digits in a byte, so we can guess that the final result will occupy about 50 GB. Most computers don't have so much memory, but 50 GB is not an unusually large disk file and the intermediate results can be stored on the disk.

The calculation requires relatively few operations. If we had $x_0 = 7^{29296875}$ we could then calculate $x_1 = x_0^2, x_2 = x_1^2, \ldots$, and then $x_{12}$ would be the answer we seek.

But we can get $x_0 = 7^{29296875}$ by first calculating $y = 7^{5859375}$, then $y' = y^2, x_0 = y'\cdot y'\cdot 7$. Done in this fashion the entire calculation of $7^{120000000000}$ requires only around 45 multiplications, of which about half are easy and half are more difficult. The difficult multiplications can be done using Fourier transform methods; the schoolbook algorithm is too slow for numbers this large.