On Waring's problem (elementary methods)
The problem in the title is to prove that for each integer \(n > 1\) there is an integer \(m > 1\) such that every integer \(N > 1\) is a sum of at most \(m\) positive \(n\)-powers. After the problem was solved for small values of \(n,\) Hilbert solved it in general. His proof uses integration. The paper reviews and simplifies proofs which do not use integration. The paper does not refer to connections with spherical designs and isometric embeddings of Banach spaces [cf., e.g., \textit{Y. I. Lyubich} and \textit{L. N. Vaserstein}, Geom. Dedicata 47, No. 3, 327--362 (1993; Zbl 0785.52002)].
scientific article; zbMATH DE number 2187244