Waring's problem: \(g(1,4)=21\) for fourth powers of positive integers (Q1825895)

The author considers a variant of Waring's problem: The problem is to determine, for given \(m\geq 0\), the exact value g(m,k) of the number \(s=s(k)\) with the property that every integer N (with an at most finite, explicitly known set of exceptions) is a sum of s k-th powers of integers \(n_ i\geq m\). Using the famous result \(g(0,4)=19\) of \textit{R. Balasubramanian}, \textit{J.-M. Deshouillers} and \textit{F. Dress} [C. R. Acad. Sci., Paris, Sér. I 303, 85-88 and 161-163 (1986; Zbl 0594.10039 and Zbl 0594.10040)] the author announces \(g(1,4)=21\) and gives the set of exceptions \(\{1,...,20\}\cup \{20+Z(1,4)\}\) where Z(1,4) is given graphically.