Displaying similar documents to “Upper bounds for k-th coset representatives modulo n”

The ternary Goldbach problem in arithmetic progressions

Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

Similarity:

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and ( N , r ) = b ∈ ℕ ³ : 1 ≤ b j ≤ r , ( b j , r ) = 1 a n d b ₁ + b ₂ + b ₃ ≡ N ( m o d r ) . It is known that    ( N , r ) = r ² ∏ p | r p | N ( ( p - 1 ) ( p - 2 ) / p ² ) ∏ 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 ≡ b j ( m o d r ) , j = 1,2,3, ⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.