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A mean value theorem on the binary Goldbach problem and its application. (English) Zbl 1188.11050

Summary: We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers \(n\) satisfying \(n \not\equiv 2 \pmod 6\), the equation \(p_{1} + p_{2} = n\) is solvable in prime variables \(p_{1}, p_{2}\) such that \(p_{1} + 2 = P_{3}\), and for every sufficiently large odd integer \(\bar n\) satisfying \(\bar n \not\equiv 1 \pmod 6\), the equation \(p_{1} + p_{2} + p_{3} = {\bar n}\) is solvable in prime variables \(p_{1}, p_{2}, p_{3}\) such that \(p_{1} + 2 = P_{2}\), \(p_{2} + 2 = P_{3}\). Here \(P_{k}\) denotes any integer with no more than \(k\) prime factors, counted according to multiplicity.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11N36 Applications of sieve methods
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[8] Tolev DI (1997) On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progresson. In: Lupanov OB (ed) Proc Steklov Inst Math 218: 414–432. Moscow: Nauka · Zbl 0911.11048
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