Incoming links: Legendre's three-square theorem
And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
4 squares are sufficient by Lagrange's four-square theorem.
3 is not enough by Legendre's three-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and