What is fastest algorithm for factoring out square from number I have $n$-digit integer $N=a^2b$, $b$ is square-free. In other words, $a$ is maximal square which divides $N$.

What is fastest known algorithm to find $a$? I can write algorithm of $O(n^2\sqrt{N})$ simply trying all squares that are smaller than $N$ and checking for divisibility.

Is this problem as hard as factoring integer?
Using state of the art factoring algorithms, you can substantially improve on the algorithm you state. It appears that no algorithm better than factoring is known at the moment. See this question on mathoverflow.