Scary Sphere

Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in three dimensional space, the Manhattan distance between those points is defined as$|x_1 – x_2| + |y_1 – y_2| + |z_1 – z_2|$.

Let $C(r)$ be a sphere with radius $r$ and center in the origin $O(0,0,0)$.
Let $I(r)$ be the set of all points with integer coordinates on the surface of $C(r)$.
Let $S(r)$ be the sum of the Manhattan distances of all elements of $I(r)$ to the origin $O$.

E.g. $S(45)=34518$.

Find $S(10^{10})$.

This problem involves a sphere in a three-dimensional (3D) space. Each point on the surface of the sphere with radius r from the origin can be represented by three variables, x, y, and z, satisfying the equation x² + y² + z² = r².

In this scenario, the points considered on the sphere must have integral values, i.e., x, y, and z are integers. So, a point P(x, y, z) lies on I(r) if x, y, and z are integers and satisfy the equation x² + y² + z² = r².

The Manhattan distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space is |x₁ – x₂| + |y₁ – y₂| + |z₁ – z₂|. In this problem, the second point is fixed at the origin O(0, 0, 0), so the equation simplifies to |x| + |y| + |z|. Thus, the Manhattan distance between a point P on the sphere and the origin is simply |x| + |y| + |z|, where P is a point on I(r).

The question asks to find S(10^10), where S(r) is the sum of Manhattan distances from I(r), i.e., the sum of |x| + |y| + |z| for all (x, y, z) in I(r). For example, S(45) = 34,518.

However, in practice, computing S(10^10) directly would be computationally demanding due to the extremely large number of integral points on the sphere with radius r = 10^10. Given the complex nature of this problem and the level of computational resources it would require, it would not actually be feasible to solve this problem by hand or with a simple calculator. You would likely need to construct a custom algorithm or computer program to efficiently solve this problem within a reasonable timeframe.

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