Goldbach's conjecture asks for with both prime. A natural attack sieves the sequence to find elements with no small prime factors, hoping to isolate primes among the . But every known sieve — Selberg's, Brun's, the general combinatorial sieve — suffers from the 'parity problem': a plain sieve's lower bound cannot distinguish a number with exactly one prime factor from one with, say, exactly three, because the sieve only 'sees' divisibility, not the parity of the number of factors. Chen's way around this obstacle was not to fight the parity problem head-on, but to relax the target: prove the weaker but still highly nontrivial statement that has at most two prime factors, for at least one prime .
The parity problem, as later formalized by Selberg, says roughly that any sieve weight built only from the Möbius function truncated at some level cannot, by itself, distinguish numbers with an odd number of prime factors below from those with an even number — since both classes contribute with the same sign pattern to . Rényi had already shown in 1947 (before the parity problem was named) that some fixed works, giving ; the sieve-theoretic content of Chen's theorem is pushing all the way down to , the sharpest value achievable by these methods to date.
- Parity problem (sieve theory)
- A fundamental obstruction, identified by Selberg, that prevents combinatorial sieve methods alone from ever proving a sequence contains infinitely many primes (numbers with exactly one prime factor), as opposed to numbers with an unspecified but small number of prime factors.