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Mathematical Notes

, Volume 88, Issue 3–4, pp 295–307 | Cite as

On the Vinogradov additive problem

  • G. I. ArkhipovEmail author
  • V. N. Chubarikov
Article
  • 80 Downloads

Abstract

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 0a, then, in the representation for k, we can set s = 0, but if k 0a − 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.

Key words

additive problem of Vinogradov Hilbert-Kamke problem Vinogradov system of equations p-solvability Waring-Goldbach problem Vinogradov system of congruences 

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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteDokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.]SteklovRussia
  2. 2.Moscow State UniversityMoscowRussia

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