##
**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 |

### Keywords:

binary Goldbach problem
Full Text:
DOI

### References:

[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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.