  Let $N$ denote a sufficiently large even integer, we define $D_{1,2}(N)$ as
the same as those in previous articles about Chen's theorem. In this paper, we
show that $D_{1,2}(N) \geqslant 1.733 \frac{C(N) N}{(\log N)^2}$, improving
previous record of Wu about $93\%$. We also get similar results on twin prime
problem and additive representations of integers. An important step in the
proof is the application of the theorems of Lichtman.
