Chen's theorem ("1 + 2")

Sieve the sequence $\mathcal{A}=\{N-p : p \text{ prime}, p\le N\}$ to detect elements with at most two prime factors, using a weighted combination of sieve sums (a Jurkat–Richert-type linear sieve) rather than a plain lower-bound sieve, since plain sieves are blocked by the parity problem. The key technical device is the 'switching principle': terms where $N-p$ has three or more prime factors $p_1p_2p_3\cdots$ are bounded above by fixing a middle-sized prime factor $p_1$ and switching to an upper-bound sieve on the primes $p \equiv N \pmod{p_1}$, which can be controlled using the Bombieri–Vinogradov theorem on primes in arithmetic progressions. Combining the resulting lower bound on the weighted count of $p+P_2$ representations with the upper bound on the unwanted $p_1p_2p_3\cdots$ contributions shows the total is positive for all large $N$. Chen announced the result in 1966 and published full details in 1973; P. M. Ross gave a simplified proof in 1975.