Primitive point packing

A point in the d-dimensional integer lattice Z d $\mathbb {Z}^d$ is primitive when its coordinates are relatively prime. Two points multiples of one another they opposite, and for this reason, we consider half within lattice, ones whose first non-zero coordinate positive. We solve packing problem that asks largest possible number such absolute values any given sum to at most a fixed k. present several consequences result intersection geometry, theory, combinatorics. In particular, obtain an explicit expression diameter zonotope contained hypercube [ 0 , k ] $[0,k]^d$ and, conjecturally polytope hypercube.