On the Vinogradov additive problem

Let us state the main result of the paper. Suppose that the collection N ₁, ..., N _n is admissible. Then, in the representation $$ \left\{ \begin{gathered} p_1 + p_2 + \cdots + p_k = N_1 , \hfill \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \hfill \\ p_1^n + p_2^n + \cdots + p_k^n = N_n , \hfill \\ \end{gathered} \right. $$ where the unknowns p ₁, p ₂, ..., p _k take prime values under the condition p _s > n+ 1, s = 1, ..., k, the number k is of the form $$ k = k_0 + b\left( n \right)s, $$ where s is a nonnegative integer. Further, if k ₀ ≥ a, then, in the representation for k, we can set s = 0, but if k ₀ ≤ a ? 1, then, for a given k ₀ there exist admissible collections (N ₁, ..., N _n ) that cannot be expressed as k ₀ summands of the required form, but can be expressed as k ₀ + b(n) summands.