Geodesic
A generalization of the Hilbert–Waring theorem
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1983), pp. 11-19 
A. A. Zenkin. A generalization of the Hilbert–Waring theorem. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1983), pp. 11-19. http://geodesic.mathdoc.fr/item/VMUMM_1983_2_a2/
@article{VMUMM_1983_2_a2,
     author = {A. A. Zenkin},
     title = {A generalization of the {Hilbert{\textendash}Waring} theorem},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {11--19},
     year = {1983},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1983_2_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $$ \mathbf Z(m,r)=\biggl\{n\mid n\neq\sum^s(n_i^r-m^r) \quad\text{for all}\quad s\geq1,n_i\geq m\biggr\} $$ and $$ \mathbf N(m,r,s)=\biggl\{n\mid n\neq\sum^s n_i^r,n_i\geq m,n> s\cdot m^r\biggr\}. $$ Then, for any $m\geq0$, $r\geq2$ there exist the number, $g(m,r)$, and the finite invariante set, $\mathbf Z(m,r)$, such that for any $s\geq g(m,r)$ $$ \mathbf N(m,r,s)=\{s\cdot m^r+z\mid z\in\mathbf Z(m,r)\}, $$ If $m=0$ then we obtain the classical Hilbert–Waring theorem.