Worked solution: Chen's weighted sieve and switching principle: every large even number is $p + P_2$ (1966/1973)

The additive representation $N = p + q$, where $p$ is prime and $q = q_1$ or $q = q_1 q_2$ has at most two prime factors (denoted $P_2$), is the deepest known unconditional approximation to Goldbach's conjecture. The standard sieve approach examines $\mathcal{A} = \{N-p : p \le N\}$. Standard combinatorial sieves assign weights $\lambda_d$ to divisors $d \mid P(z)$, where $P(z) = \prod_{p < z} p$, to sift out multiples of small primes below $z$. However, Selberg's parity phenomenon presents a fundamental barrier: sieve weights based solely on Möbius truncations cannot distinguish integers with an odd number of prime factors from those with an even number. For a linear sieve of dimension 1 with level of distribution $D$, the optimal lower-bound function $f(s)$, where $s = \log D / \log z$, vanishes identically for $s \le 2$. Setting $z = N^{1/2}$ to isolate primes among $N-p$ forces $s = 1$ when $D = N^{1/2}$, causing the sieve lower bound to vanish completely.

Because of this obstacle, mathematicians pursued almost-prime approximations $N = p + P_K$. Alfréd Rényi (1947) first established $N = p + P_K$ for some large constant $K$ using early large sieve ideas. Theodor Estermann proved $N = p + P_6$ under GRH in 1932, and Wang Yuan proved $N = p + P_4$ in 1956 and $N = p + P_3$ in 1962 under GRH using Kuhn's weighted sieve. The breakthrough came with the Bombieri–Vinogradov theorem (1965), which unconditionally established the average level of distribution $D = N^{1/2 - \varepsilon}$ for primes in progressions, allowing A. I. Vinogradov, M. B. Barban, and Pan Chengdong to make $N = p + P_3$ unconditional.

Reaching $K = 2$ seemed blocked by parity, because linear sieves could not eliminate candidates with three prime factors without destroying the lower bound. Chen Jingrun circumvented this obstacle not by attempting to break the parity barrier directly, but by constructing a weighted detector for $\Omega(N-p) \le 2$. Instead of relying solely on a lower-bound sieve, Chen combined a lower-bound linear sieve with an ingenious upper-bound count on unwanted three-factor exceptions, completely bypassing the parity obstruction.