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The ternary Goldbach problem.

David Rodney (Roger) Heath-Brown (1985)

Revista Matemática Iberoamericana

The object of this paper is to present new proofs of the classical ternary theorems of additive prime number theory. Of these the best known is Vinogradov's result on the representation of odd numbers as the sums of three primes; other results will be discussed later. Earlier treatments of these problems used the Hardy-Littlewood circle method, and are highly analytical. In contrast, the method we use here is a (technically) elementary deduction from the Siegel-Walfisz Prime Number Theory. It uses...

The ternary Goldbach problem in arithmetic progressions

Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

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.

The value of additive forms at prime arguments

Roger J. Cook (2001)

Journal de théorie des nombres de Bordeaux

Let f ( 𝐩 ) be an additive form of degree k with s prime variables p 1 , p 2 , ⋯ , p s . Suppose that f has real coefficients λ i with at least one ratio λ i / λ j algebraic and irrational. If s is large enough then f takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for k ≥ 6 , Heath-Brown’s improvement on Hua’s Lemma.

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