Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a necessary and(or) sufficient condition for $n(n-1)/2$ integers to be $n$ points distances?
For example can the distances be $n(n-1)/2$ consecutive integers? Can all the distances be prime numbers?
One constraint on distances between points in a plane is that for any four points the Cayley-Menger determinant $$ \det \pmatrix{0 & 1 & 1 & 1 & 1\cr 1 & 0 & d_{12}^2 & d_{13}^2 &d_{14}^2\cr 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2\cr 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2\cr 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0\cr} = 0$$ For example, the only case where the six consecutive integers $d_0,d_0+1,\ldots, d_0+5$ can be the distances between four coplanar points is $d_0 = 1$, where the four points are collinear, e.g. at $(0,0), (1,0), (4,0), (6,0)$.
