For every k, we denote the least such s by g(k).
[Lagranges Theorem]? states that every natural number is the sum of at least four squares; since three squares are not enough, this theorem establishes g(2)=4. g(3)=9 was established around 1912 and g(4) = 19 in 1986. These values had already been conjectured by Waring.
Using the [Hardy Littlewood Method]?, g(k) can now readily be computed for all other values of k as well.