Lagrange's four square theorem states that any natural number $n$ can be written as the sum of the square of 4 other integers. For most values of $n$, there are multiple square combinations that work. For example, $16=4^2$ and also $16=2^2 + 2^2 + 2^2 + 2^2$. Is there a name for the solution where first term is as large as possible, the second term is as large as possible (given the value of the first term), and so on? For 16, this would be the $4^2$ solution. I'd still want no more than 4 nonzero terms.
Has this solution been discussed anywhere? I like this solution because it's unique.
Also, would that solution be equivalent to the solution with the fewest nonzero terms?
If no name for this solution exists, what name would you suggest? Names that occur to me are the minimum entropy solution, or the maximum bias solution.