Distributions of the values of some arithmetical functions

Paul Erdös; Andrzej Schinzel

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M. B. S. Laporta (1997)

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Given A and B integers relatively prime, we prove that almost all integers n in an interval of the form [N, N+H], where N exp(1/3+e) ≤ H ≤ N can be written as a sum Ap1 + Bp2 = n, with p1 and p2 primes and e an arbitrary positive constant. This generalizes the results of Perelli et al. (1985) established in the classical case A=B=1 (Goldbach's problem).

The ternary Goldbach problem in arithmetic progressions

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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N, r) = b \in ℕ ³ : 1 \leq b_{j} \leq r, (b_{j}, r) = 1 a n d b ₁ + b ₂ + b ₃ \equiv N (m o d r) .$ It is known that $(N, r) = r ² \prod_{p | r} p | N ((p - 1) (p - 2) / p ²) \prod_{p | r} p ∤ N ((p ² - 3 p + 3) / p ²)$ . Let ε > 0 be arbitrary and $R = N^{1 / 8 - ε}$ . We prove that for all positive integers r ≤ R, with at most $O (R l o g^{- A} N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_{j} \equiv b_{j} (m o d r),$ j = 1,2,3,⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.