How to find the closest N to the power of X to the given number?
Here is a general solution for the following problem:


  Given a positive integer $m$, find positive integers $a,b \geq 2$ such that $a^b$ is as close as possible to $m$.


Let $n$ be the length of $m$ in bits (so $n = \Theta(\log m)$). If $2^b &gt; m$ then there is no point to check any $b_0 &gt; b$, hence we only need to check $O(\log m) = O(n)$ many different values of $b$ (we can find the maximal $b$ by comparing $2^b$ to $m$). For each value of $b$, we can find the best value of $a$ using binary search; this takes polynomial time for each $b$, and so polynomial time overall. Finally, we choose the optimal solution among the $O(\log m)$ possibilities.