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

Step 2 of 4: A weighted linear sieve, following Jurkat–Richert

In plain words

Instead of a single sieve sum, Chen combines several: a main sieve sum $S(\mathcal{A},\mathcal{P},z)$ counting shifted primes $N-p$ with no prime factor below $z\approx N^{1/10}$ (candidates for $q=1$ or $q$ prime or $q=P_2$ ), together with correction sums that subtract off the unwanted cases where $N-p$ has three or more prime factors below the cutoff. The Jurkat–Richert theorem (1965) supplies sharp, explicit lower and upper bounds for such sieve sums in terms of two universal functions $F_1,f_1$ solving a linear differential-difference system, which is exactly the technology needed to make the combination provably positive.

S(\mathcal{A},\mathcal{P},z) = \#\{n\in\mathcal{A} : p\mid n \Rightarrow p\ge z\},\qquad \mathcal{A}=\{N-p : p\le N\}

Detailed analysis

Concretely, Chen studies a combination such as $W(N)=\sum_{\substack{N-p\,\text{squarefree}\\p\le N}} \Big(1-\tfrac12\sum_{\substack{N-p=p_1p_2p_3\\ z\le p_1<p_2<p_3}}1\Big)$ , so that $W(N)>0$ certifies the existence of a prime $p\le N$ for which $N-p$ has no factor below $z$ and is not a product of three primes all above $z$ — forcing $N-p$ to be prime or a $P_2$ (any remaining prime factor below $z$ can only appear with multiplicity giving at most two total factors, given the cutoff $z\approx N^{1/10}$ ). The first sum is a standard sieve, boundable below by Jurkat–Richert; the correction sum is precisely where the switching principle of the next step is needed.