how to quickly determine whether a given natural number is a power of any other natural number? - MathOverflow

how to quickly determine whether a given natural number is a power of any other natural number?

2010-02-02T18:06:08Z

We have a natural number $n>1$. We want to determine whether exists any natural numbers $a, k>1$ such that $n = a^k$.

Please suggest a polynomial-time algorithm.

Answer by Felipe Voloch for how to quickly determine whether a given natural number is a power of any other natural number?

2010-02-02T18:26:17Z

For each $k \le \log n/\log 2$, compute an approximation to the positive real $k$-th root of $n$ using Newton's method to enough precision to check if it is an integer. Alternatively, use $p$-adic roots for a suitable $p$, with Newton turning into Hensel.

Answer by Ben Linowitz for how to quickly determine whether a given natural number is a power of any other natural number?

2010-02-02T18:40:06Z

In order to test whether or not a natural number $n$ is a perfect power, we can conduct a binary search of the integers {1,2,...,n} for a number a number $m$ such that $n = m^b$ for some $b>1$. Let $b>1$. If a solution $m$ to $m^b =n$ exists,then it must lie in some interval $[c_i,d_i]$. When $i = 0$ we may take $[c_0,d_0] = [1,n]$. To define $[c_{i+1},d_{i+1}]$, consider $\alpha:= \left\lfloor \frac{(ci+di)}{2}\right\rfloor$. If $\alpha^b = n$ then we’re done. If $\alpha^b > n$, let $[c_{i+1}, d_{i+1}] = [c_i, \alpha]$; otherwise $\alpha^b < n$ and we let $[c_{i+1}, d_{i+1}] = [\alpha, d_i]$. We continue in this manner until $|c_i − d_i| \leq 1$. We then increase the value stored in variable $b$ and start the loop again. Performing this loop for all $b \leq log(n)$ completes the algorithm.

A pseudocode implementation of this algorithm can be found on page 21 of Dietzelbinger's Primality Testing in Polynoial Time. Its complexity is about $O(log^3(n))$.

Answer by Pace Nielsen for how to quickly determine whether a given natural number is a power of any other natural number?

2010-02-02T19:03:04Z

This can be done in "essentially linear time." Check out Daniel Bernstein's website: http://cr.yp.to/arith.html

Especially note his papers labeled [powers] and [powers2].