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

Frequently asked questions

Why doesn't this method extend from

P_2

down to a genuine prime, proving Goldbach's conjecture itself?

The weighted sieve inequality only ever proves a lower bound is positive; it cannot certify that the surviving $N-p$ has exactly one prime factor rather than exactly two, precisely because that distinction is the parity problem the sieve cannot see. Reaching genuine primes would need a fundamentally different tool that goes beyond sieve weights alone — no such tool is currently known.

Source

Chen Jingrun (1966 announcement, 1973 full proof); simplified by P. M. Ross (1975), 1973

On the representation of a larger even integer as the sum of a prime and the product of at most two primes

Condensed summary

Step 3 of 4: The switching principle: trading roles between

p

and

p_1

Previous step Next step

In plain words

Bounding the correction sum directly — over all primes $p\le N$ and all factorizations $N-p=p_1p_2p_3$ — mixes two different variables ( $p$ and $p_1$ ) in a way ordinary sieve bounds cannot handle efficiently. Chen's switching principle changes perspective: for a fixed prime $p_1$ in a suitable middle range, the condition ' $p_1$ divides $N-p$ ' is just the congruence $p\equiv N\pmod{p_1}$ . So instead of sieving $N-p$ over varying $p$ , one sieves the primes $p$ themselves inside a fixed residue class mod $p_1$ — trading the original hard 'three-factor' sieve problem for an easier problem about the distribution of primes in arithmetic progressions, summed over $p_1$ .

N-p = p_1p_2p_3 \ (p_1<p_2<p_3) \ \ \xrightarrow{\text{switch}} \ \ p \equiv N \!\!\pmod{p_1}, \quad p_1 \ \text{fixed in a middle range}

Detailed analysis

After switching, bounding the correction sum reduces to bounding $\sum_{p_1} \pi(N; p_1, N)$ — counts of primes $p\le N$ in the residue class $N \bmod p_1$ — uniformly over primes $p_1$ up to roughly $N^{1/3}$ . This is exactly the regime where the Bombieri–Vinogradov theorem applies: it shows that, on average over moduli $q$ up to $N^{1/2-\varepsilon}$ , the count of primes in any residue class mod $q$ is as expected from the prime number theorem, with an error small enough (unconditionally, with no need for the still-unproved Generalized Riemann Hypothesis) to make the correction sum provably smaller than the main term from Step 2. Combining the two gives $W(N)>0$ for all sufficiently large even $N$ .