Can sieve method prove ternary (three) prime Goldbach conjecture?

Question

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ?

I had done some research, I could not find any articles on this. Can anyone provide some help on this ?

I assume that sieve method will be prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem), because it can prove (1, 2) Chen's Theorem.

Answer 1

This is called the weak Goldbach conjecture http://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture
and was apparently solved by Harald Helfgott last year. I don't know what his methods were.

Answer 2

Terrence Tao writes that it would be difficult to prove the weak Goldbach conjecture due to the parity problem. Most of the progress has been made by a combination of the circle method and Fourier techniques.