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  "body": "\n## Statement\n\nCall a positive integer a *$P_2$ number* if it is a [[prime|01826KMG2H]] or a product of two primes.\nEquivalently, its [[number of prime factors counted with multiplicity|0KXQKES075]] is $1$ or $2$.\n\n1. There is an integer $N_0$ such that every even integer $N \u003e N_0$ can be written as $N = p + m$ with $p$ prime and $m$ a $P_2$ number.\n2. There are infinitely many primes $p$ such that $p + 2$ is a $P_2$ number.\n\n## Remarks\n\nBoth parts are due to Chen Jingrun.\nThe proof is a weighted sieve argument.\nA lower bound sieve counts the $m = N - p$ with no small prime factor, and an upper bound sieve removes those with three or more prime factors.\nThe [[Bombieri–Vinogradov theorem|B3JMY54GJF]] controls the remainder terms on average over the moduli.\nA switching argument, which counts the same configurations with the roles of the factors exchanged, is the key new step.\nExample: $100 = 3 + 97 = 7 + 93$, where $97$ is prime and $93 = 3 \\cdot 31$ is a $P_2$ number.\nPart 1 is a weak form of [[Goldbach's conjecture|063JNBSMJ0]], and part 2 a weak form of the [[twin prime conjecture|0FER36N8SB]].\n\n## Explanations\n\n### L1 human\n\nGoldbach's conjecture says that every even number larger than $2$ is a sum of two primes.\nChen proved the next best thing: every large even number is a prime plus a number that is either prime or the product of two primes.\nIn the same way, he proved that there are infinitely many primes $p$ for which $p + 2$ is \"almost prime\".\n\n### L2 human\n\nSieve methods cannot by themselves separate numbers with an odd number of prime factors from those with an even number; this is the parity problem.\nSo a sieve can at best reach products of two primes, and Chen's theorem reaches exactly that limit.\nEarlier sieve results of the same kind, starting with the [[Brun sieve|M4G4RXASTY]], allowed more prime factors on both sides.\n",
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