Geodesic
About simultaneous representation of numbers by sum of primes
Čebyševskij sbornik, Tome 13 (2012) no. 2, pp. 12-17.

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In this paper proved theorem Theorema. If $X$ -it is enough big, $\delta$ ($0\delta1$) it is enough small real numbers, that fair estimation $$ J(\overrightarrow{b})>\frac{\Bigl(\frac{1}{\sqrt{n}}3(n!)^{2}B^{(2n-1)}|\overrightarrow{b}|\Bigr)^{1-\frac{\delta}{10(n-1)}}}{\Bigl(\ln\Bigl(\frac{1}{\sqrt{n}}3(n!)^{2}B^{(2n-1)}|\overrightarrow{b}|\Bigr)\Bigr)^{n+1}}, $$ for all vector $\overrightarrow{b}\in U(X)$ with the exclusion of no more than $$ E(X)^{n-\frac{\delta}{17n^{3}}} $$ the vector of them. Here $B=\max\{3|a_{ij}|\}$, $1\leq i \leq n$, $1\leq j \leq n+1$. 
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I. Allakov; A. Safarov. About simultaneous representation of  numbers by sum of primes. Čebyševskij sbornik, Tome 13 (2012) no. 2, pp. 12-17. http://geodesic.mathdoc.fr/item/CHEB_2012_13_2_a2/

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