Project Leonhard

Number of Lattice Points in a Hyperball

Problem Statement

Let $T(r)$ be the number of integer quadruplets $x, y, z, t$ such that $x^2 + y^2 + z^2 + t^2 \le r^2$. In other words, $T(r)$ is the number of lattice points in the four-dimensional hyperball of radius $r$.

You are given that $T(2) = 89$, $T(5) = 3121$, $T(100) = 493490641$ and $T(10^4) = 49348022079085897$.

Find $T(2^N)$.

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You need to submit in the format: "N:problem(N)", possibly with multiple values at once, separated by commas, with $N$ between $1$ and $80$.

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