For this puzzle, the only valid triangles are those whose three side lengths are prime numbers p,q,r (not necessarily distinct).

The task is to find valid triangles embedded in a given lattice; in other words, triangles whose three vertices are lattice points.

An infinite family of lattices is available: for each positive integer n there is a "n-dimensional integer lattice", denoted as Z^n, which is the set of all points with integer coordinates in n-dimensional Euclidean space.

Euclidean distance between lattice points A=(a_1,...a_n) and B=(b_1,...,b_n) is computed as usual:
d(A,B) = sqrt((a_1-b_1)^2 + ... + (a_n-b_n)^2).

The vertices of a proper triangle do not all lie on the same straight line, so lattice Z^1 must be discarded and the first few lattices to be considered are the "square lattice" Z^2 on the plane and the "cubic lattice" Z^3 in 3D space.

Answer the following questions.

Q1) Which valid triangles are embeddable in Z^2?
Q2) Which ones are embeddable in Z^3 but not in Z^2?
Q3) Which ones are embeddable in Z^4 but not in Z^3?

Q4) Which valid equilateral triangles are embeddable in some lattice Z^n?
Q5) Which valid triangles are embeddable in no lattice Z^n, no matter which value of n is chosen?
Q6) Is there some upper bound nmax such that any valid triangle embeddable in some lattice Z^n is also guaranteed to be embeddable in Z^nmax?

Some (non-valid) examples.
The rectangular triangle with side lengths 3,4,5 is easily embeddable in Z^2, by choosing vertices:
{(0,0),(4,0),(0,3)}.
The isosceles triangle with side lengths 4,7,7 is not embeddable in Z^2 but it's embeddable in Z^3, by choosing vertices:
{(0,0,0),(4,0,0),(2,3,6)}.

I hope that you'll find this puzzle interesting. It was partially inspired by puzzle 1155, also related to triangle searching and prime distances.