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### The ternary Goldbach problem in arithmetic progressions

Acta Arithmetica

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $\left(N,r\right)=b\in ℕ³:1\le {b}_{j}\le r,\left({b}_{j},r\right)=1andb₁+b₂+b₃\equiv N\left(modr\right).$It is known that    $\left(N,r\right)=r²{\prod }_{p|r}p|N\left(\left(p-1\right)\left(p-2\right)/p²\right){\prod }_{p|r}p\nmid N\left(\left(p²-3p+3\right)/p²\right)$. 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.

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### On the almost Goldbach problem of Linnik

Journal de théorie des nombres de Bordeaux

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 $2$.

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