How to find integer nth roots?

Question

I want to find the greatest integer less than or equal to the kth root of n. I tried

int(n**(1/k))

But for n=125, k=3 this gives the wrong answer! I happen to know that 5 cubed is 125.

>>> int(125**(1/3))
4

What's a better algorithm?

Background: In 2011, this slip-up cost me beating Google Code Jam problem Expensive Dinner.

Answer 1

One solution first brackets the answer between lo and hi by repeatedly multiplying hi by 2 until n is between lo and hi, then uses binary search to compute the exact answer:

def iroot(k, n):
    hi = 1
    while pow(hi, k) < n:
        hi *= 2
    lo = hi // 2
    while hi - lo > 1:
        mid = (lo + hi) // 2
        midToK = pow(mid, k)
        if midToK < n:
            lo = mid
        elif n < midToK:
            hi = mid
        else:
            return mid
    if pow(hi, k) == n:
        return hi
    else:
        return lo

A different solution uses Newton's method, which works perfectly well on integers:

def iroot(k, n):
    u, s = n, n+1
    while u < s:
        s = u
        t = (k-1) * s + n // pow(s, k-1)
        u = t // k
    return s