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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r)=b\in \mathbb{N}³:1\le b_j\le r,(b_j,r)=1andb₁+b₂+b₃\equiv N\left(modr\right).$It is known that    $(N,r)=r²{\prod }_{p|r}p|N((p-1)(p-2)/p²){\prod }_{p|r}p\nmid N((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R=N^{1/8-\epsilon }$. We prove that for all positive integers r ≤ R, with at most $O\left(Rlo{g}^{-A}N\right)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j\equiv b_j\left(modr\right),$ j = 1,2,3,$$⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

Under the Generalized Riemann Hypothesis, it is proved that for any $k\ge 200$ there is $N_k\>0$ depending on $k$ only such that every even integer $\ge N_k$ is a sum of two odd primes and $k$ powers of $2$.