Given the radix-$r$ representation of a integer $n$, and a small integer constant $k$, there is an $O(\log n)$ algorithm for detecting whether $n$ is a multiple of $k$, namely, division, which produces as a byproduct the quotient $\lfloor n/k\rfloor$. In general this is the best one can do. But for certain choices of $r$ and $k$, for example $r=10$ and $k=2$, there is an algorithm which answers the question much faster (constant time) without producing the quotient.
Given the radix-$r$ representation of a integer $n$, we can extract the integer square root $\lfloor\sqrt n\rfloor$ in something like $O(\log^3 n)$ time by doing binary search, which Joriki notes below can be improved to $O(\log^2 n)$ with a sufficiently clever implementation. This gives an $O(\log^2 n)$ algorithm for determining whether $n$ is a perfect square.
Is there a significantly faster algorithm which correctly decides whether $n$ is perfect square, without also producing the square root? I suspect not, but I would be interested to see a proof.
Best Answer
See the paper by Bernstein, Lenstra, and Pila: Detecting Perfect Powers by Factoring into Coprimes, Mathematics of Computation, Volume 76, #257, January 2007, pp. 385-388. Or here.
From the abstract: This paper presents an algorithm that, given an integer n>1, finds the largest k such that n is a kth power.
The algorithm runs in time $\log(n)(\log\log(n))^{O(1)}$.