Ciro Santilli 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
Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
In other words, it is unlikely to have a Conjecture reduction to a halting problem.
It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.