On the simultaneous representation of numbers by the sum of five prime numbers

Let $X-$be a sufficiently large real number, $b_{1},b_{2},b_{3}-$be integers with the condition $1\le {{b}_{1}},{{b}_{2}},{{b}_{3}}\le X,\,\,\, a_{ij}, (i=1,2,3;\,\,\, j=\overline{1.5})$ positive integers, $p_{1},...,p_{5}-$prime numbers. Let us set $B=max\{3|a_{ij}|\} ,\,\,(i=1,2,3;\,\,j=\overline{1.5}), \vec{b} = (b_{1},b_{2},b_{3}),\,\, K=36\sqrt{3}B^{5}|\vec{b}|, E_{3,5}(X)=\\=card\{b_{i} |1\le {{b}_{i}}\le X,\,\,b_{i}\neq a_{i1} p_{1}+\cdots+a_{i5} p_{5},\,\,i=1,2,3\}$. In the paper it is proved that the system $b_{i}=a_{i1}p_{1}+\cdots+a_{i5}p_{5},\,\,(i=1,2,3)$ is solvable in prime numbers $p_{1},\cdots,p_{5}$, for all triples $\vec{b}=(b_{1}, b_{2},b_{3}),\,\, 1\le {{b}_{1}},{{b}_{2}},{{b}_{3}}\le X$, with the exception of no more than $E_{3,5}(X)$ triples of them, and a lower bound is obtained for the $R(\vec{b})-$number of solutions of this system, that is, the inequality $R(\vec{b})>> K^{2-\varepsilon}( \log K)^{-5}$ is proved to be true, for all $(b_{1},b_{2},b_{3})$ with the exception of no more than $X^{3-\varepsilon}$ triples of them.

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