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The ternary Goldbach problem in arithmetic progressions

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Liu J., Zhan T.

Acta Arithmetica

1997

tom 82

nr 3

197-227

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $𝓑(N,r) = {b∈ ℕ³ : 1 ≤ b_j ≤ r, (b_j, r) = 1$ and b₁+b₂+b₃ ≡ N (mod r)}. It is known that $#𝓑(N,r) = r² ∏_{p|r}_{p|N} ((p-1)(p-2)/p²) ∏_{p|r}_{p∤N} ((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R = N^{1/8-ε}$. We prove that for all positive integers r ≤ R, with at most $O(Rlog^{-A}N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j ≡ b_j (mod r),$ j = 1,2,3,$ ⎩ with prime variables is solvable whenever b ∈ 𝓑(N,r), where A > 0 is arbitrary.

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1 Acta Arithmetica

1 Liu J.

1 Zhan T.

1 1997

Wyniki wyszukiwania

The ternary Goldbach problem in arithmetic progressions