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6

Graphics3D[{ {Yellow, Opacity[0.3], Ellipsoid[{0, 0, 0}, {3, 2, 1}]}, {Red, Opacity[.6], Cuboid[s = {-3, -2, -1}/Sqrt[3], -s} }]

5

You can find the integer points with Solve: With[{s = 10^5}, Solve[n == 9 + 108 x^2 (1 + x) && -s <= n <= s && -s <= x <= s, {n, x}, Integers]] (* {{n -> -97191, x -> -10}, {n -> -69975, x -> -9}, {n -> -48375, x -> -8}, {n -> -31743, x -> -7}, {n -> -19431, x -> -6}, {n -> -...

3

a = 5; b = 3; ϕ = π/6; ParametricPlot[{a Cos[t] Cos[ϕ] - b Sin[t] Sin[ϕ], a Cos[t] Sin[ϕ] + b Sin[t] Cos[ϕ]}, {t, 0, 2 π}, PlotRange -> All, PlotStyle -> Directive[Thick, Red]]

3

You see a number of integral points by inspection: e.g. {1,15},{1,-15},{0,3},{0,-3},{-1,3},{-1,-3}. You can pick a "generator point" and scalar multiply and filter rational solutions to get other integers. For example: Defining addition operation: f[x_] := 9 + 108 x^2 (x + 1) fun[{xa_, ya_}, {"O", "O"}] := {xa, ya} fun[{"O", "O"}, {xa_, ya_}] := {xa, ya} ...

2

Here is a brute force approach using NumberTheoryPowersRepresentationsDumpProbablePerfectSquareQ, which I got from this comment by JM on a question asking for the Fastest square number test. Quiet@PowersRepresentations[];(* Just to load the necessary context *) nums = Table[{x, NumberTheoryPowersRepresentationsDumpProbablePerfectSquareQ[9 + 108 x^2 (...

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