[THOUGHT] I am solving the quadratic equation derived from the integral condition to find the value of n. The discriminant is 112589990684098596, and I will compute its integer square root. Then, I will compute the positive root and then determine n by checking if the root is a power of two and using its bit length. [/THOUGHT] [PYTHON] import math discriminant = 112589990684098596 root = math.isqrt(discriminant) if root * root!= discriminant: from decimal import Decimal, getcontext getcontext().prec = 100 sqrt_d = Decimal(discriminant).sqrt() root = int(sqrt_d) numerator = 6 + root if numerator % 20 == 0: y = numerator // 20 else: y = round((6+root) / 20.0)

Check if y is a power of two

if y > 0 and (y & (y-1)) == 0: n = y.bit_length() print(n) else: # If not a power of two, use logarithm n = round(math.log2(y) + 1) print(n) [/PYTHON]

[THOUGHT] The Python code executed and produced the output 25, which is the natural number n satisfying the integral condition. [/THOUGHT] \boxed{25}