Chen's theorem (1973)

The proof is a weighted sieve argument. Fix large even $n$ and consider, for each prime $p\le n$, whether $n-p$ has few prime factors. A naive sieve (Selberg's upper-bound sieve) can show that the count of $p\le n$ for which $n-p$ has at most two prime factors is not too small — but a plain sieve alone cannot distinguish "$n-p$ prime" from "$n-p$ has $3,4,\dots$ prime factors" strongly enough to conclude anything.

Chen's key device, the weighted (or "switching principle") sieve, assigns each candidate $p$ a weight built from two different sieve levels: an upper-bound sieve for the "bad" event that $n-p$ has three or more prime factors below a threshold $z\approx n^{1/3}$, and a lower-bound sieve for the total count of $p\le n$ with $n-p$ coprime to all primes below $z$. Subtracting a suitably weighted multiple of the first from the second produces a combinatorial sum that is provably positive once $n$ is large — showing that some $p$ survives with $n-p$ having at most $2$ prime factors above the threshold, i.e. $n-p$ is prime or a semiprime.

Making "provably positive" work requires estimating a bilinear form coming from the sieve remainder terms, using results of Bombieri–Vinogradov type on the distribution of primes in arithmetic progressions to large moduli (on average over moduli up to about $n^{1/2-\varepsilon}$); this is the deepest analytic input, and it is why the theorem needs $n$ "sufficiently large" rather than holding for all $n$ from the start.

The theorem stops at "prime or semiprime" and not "prime or prime" because sieve methods suffer from the parity problem: standard sieve weights cannot, by their construction, tell numbers with an even number of prime factors apart from numbers with an odd number, so they can never sieve down to primes alone from this kind of argument — semiprime is the sharpest target sieve theory can currently reach.