Is there a polynomial time algorithm to test whether a number exponent of some number?

Question

Just study the famous paper PRIMES is in P and get confused.

First step of the proposed algorithm is If (n=a^b for nature number a and b>1), output COMPOSITE. Since the whole algorithm runs in polynomial time, this step must also complete in O((log n)^c)(given input size is O(log n). However, I can't figure out any algorithm to hit the target after some googling.

QUESTION:

Is there any algorithm available to test whether a number exponent of some other number in polynomial time?

Thanks and Best Regards!

Answer 1

If n=a^b (for a > 1) then b ≤ log₂ n, we can check for all b's smaller than log n to test this, we can iterate for finding b from 2 to log n, and for finding a we should do binary search between 1..sqrt(n). But binary search takes O(logn) time for iteration, finally in each step of search(for any found a for checking) we should check that whether a^b == n and this takes O(log n), so total search time will be O(log³n). may be there is a faster way but by knowing that AKS is O(log⁶n) this O(log³n) doesn't harm anything.