Python Language Computing large integer roots


Example

Even though Python natively supports big integers, taking the nth root of very large numbers can fail in Python.

x = 2 ** 100
cube = x ** 3
root = cube ** (1.0 / 3)

OverflowError: long int too large to convert to float

When dealing with such large integers, you will need to use a custom function to compute the nth root of a number.

def nth_root(x, n):
    # Start with some reasonable bounds around the nth root.
    upper_bound = 1
    while upper_bound ** n <= x:
        upper_bound *= 2
    lower_bound = upper_bound // 2
    # Keep searching for a better result as long as the bounds make sense.
    while lower_bound < upper_bound:
        mid = (lower_bound + upper_bound) // 2
        mid_nth = mid ** n
        if lower_bound < mid and mid_nth < x:
            lower_bound = mid
        elif upper_bound > mid and mid_nth > x:
            upper_bound = mid
        else:
            # Found perfect nth root.
            return mid
    return mid + 1

x = 2 ** 100
cube = x ** 3
root = nth_root(cube, 3)
x == root
# True