Goldbach-Waring problem

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A problem concerning the behaviour of the number of solutions $I_k(N)$ of the equation


where $p_1,\dotsc,p_k$ are prime numbers and $n\geq1$ (cf. Waring problem; Goldbach problem). The results obtained (up to 1977) are roughly the same as those obtained in Waring's problem: It has been proved that the equation is solvable (i.e. $I_k(N)>0$) if $k=O(n\log n)$, while an asymptotic formula for $I_k(N)$ has been obtained for $k=O(n^2\log n)$. These solutions were obtained by the Vinogradov method.


[1] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[2] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)
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