Does there exist an unbounded convex polyhedron with faces that have 3, 5, 7, 11, 13, ...
edges, i.e., such that the number of edges of each face realize exactly the odd primes, with each prime realized exactly once (i.e., there is just one face with that number of edges)? If so, can this be achieved with adjacent primes realized by adjacent faces?
Here is a start. :-)
Update. Now answered negatively by Ed Pegg! For a related question, which does have a positive solution, see the MO question, "Polyhedra that combinatorially shadow a sequence": There is a polyhedron whose shadows combinatorially mimic the primes when the polyhedron is appropriately continuously reoriented.