p-adic square root approximation

import math
import matplotlib.pyplot as plt

import fractions

def p_valuation(p, n):

    if n == 0:
        return math.inf

    result = 0

    while n % p == 0:   # Note that "%" is the modulo operation
        n = n / p
        result += 1

    return result


def p_norm(p, z):
    return p ** (p_valuation(p, z.denominator) - p_valuation(p, z.numerator))   # Note that "**" is exponentiation


def one_plus_square_approximate(x, N):

    s = 1

    for i in range(1, N + 1):
        numerator = (-1)**(i + 1) * math.factorial(2 * i) * (x ** i)
        denominator = (4 ** i) * (math.factorial(i) ** 2) * (2 * i - 1)

        s += fractions.Fraction(numerator, denominator)

    return s


for N in range(1, 10):
    approximation = int(one_plus_square_approximate(-8, N))
    square = approximation ** 2
    diff = square + 7
    diff_norm = p_norm(2, diff)

    print("{:15d} &  {:15d} &  {:15d} &  {:15d} &  {:15f} \\\\".format(N, approximation, square, diff, diff_norm))