consider a sphere centered at origin and radius $m \in \mathbb{N}.$ Let $S$ be the set of all points in or on the sphere.Is it possible to find the number of traingles with vertices in $S?$What about the number of traingles with vertices in $S$ and integer side lengths?I posed this problem to myself but I have no idea where to start?What about a the generilisation of the problem in $n-$dimensional space?I would be highly obliged and relieved if somebody could provide any hints/suggestions or any links if the problem has already been considered in the literature .
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$\begingroup$ The earlier MO question, Exactly Counting the Number of Lattice Points in an š¯‘›-Dimensional Sphere, addresses some parts of your question. $\endgroup$– Joseph O'RourkeCommented Sep 15, 2022 at 22:22
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1$\begingroup$ Does this answer your question? Integral triangles in a hypersphere $\endgroup$– D.W.Commented Oct 1, 2022 at 5:40
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