Calculate Nth root with integer arithmetic

Question

There are a couple of ways to find integer square roots using only integer arithmetic. For example this one. It makes for interesting reading and also a very interesting the

Answer 1

You can use Newton's method using only integer arithmetic, the step is the same as for floating point arithmetic, except you have to replace floating point operators with the corresponding integer operators in languages which have different operators for these.

Let's say you want to find the integer-k-th root of a > 0, which should be the largest integer r such that r^k <= a. You start with any positive integer (of course a good starting point helps).

int_type step(int_type k, int_type a, int_type x) {
    return ((k-1)*x + a/x^(k-1))/k;
}

int_type root(int_type k, int_type a) {
    int_type x = 1, y = step(k,a,x);
    do {
        x = y;
        y = step(k,a,x);
    }while(y < x);
    return x;
}

Except for the very first step, you have x == r <==> step(k,a,x) >= x.