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Problem #418

Into the 4th dimension
 Public ★(x14) 09/01/17 by Philippe_57721 5xp Math 66.7%

Consider a sphere whose square of its radius is $n$ and centered at $(0,0,0,0)$ in the $4^{th}$ dimension space.
For a given $n$, the sphere touches a certain number of integer lattice points.

For instance, if $n = 7$, it touches the point (1,2,1,1)

Let $t(n)$ the number of integer lattice points touched by the sphere with square of radius $n$

Find $\sum_{k=0}^{100}t(10^{17}+k)$

You are given $\sum_{k=0}^{100}t(1000+k)=1042664$

[My timing: 9 sec]

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