Chen’s theorem

Theorem. Every sufficiently large even integer $n$ can be expressed as the sum of two primes $p+q$, or the sum of a prime and a semiprime $p+qr$, where $p$, $q$ and $r$ are all distinct primes. “Sufficiently large” could mean $n>60$. For example, 62 can be represented as $p+qr$ in seven different ways: $5+3\times 19$, $7+5\times 11$, $11+3\times 17$, etc.

This theorem was proven by Chen Jingrun in 1966 but had to delay publishing his results until 1973 because of political problems in his native China. Chen’s proof has been considered “a highly technical application of sieving methods.” (Eisenstein et al, 2004) Ross simplified Chen’s proof almost a decade later. Still, a summary of the proof can run for dozens of pages. A much shorter, but excessively broad summary that can fit in here goes something like this: reduction to sieving, estimation of sieving functions, search for upper bounds using the Jurkat-Richert theorem, using a bilinear form inequality, and joining together of all these results to create a function that counts the number of representations of a given number as either $p+q$ or $p+qr$, and showing that that function always returns a positive integer when the given number is sufficiently large.

The Zumkeller-Lebl conjecture, an attempt to generalize Chen’s theorem to odd numbers, and unproven as of 2008, states that sufficiently large integers, be they even or odd, can also be represented as $p+qr$. To represent odd numbers this way, only one of the primes can be 2 (or both $q=r=2$). Levy’s conjecture, which applies only to odd numbers, has $q=2$ and $p$ and $r$ both odd primes.

Sequence A100952 of Sloane’s OEIS lists the known twenty-one small integers that can’t be expressed as specified by the theorem.