Chen's theorem

Variations

Chen's 1973 paper stated two results with nearly identical proofs. His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the twin prime conjecture. It states that if h is a positive even integer, there are infinitely many primes p such that p + h is either prime or the product of two primes.

Ying Chun Cai proved the following in 2002:

There exists a natural number ${\displaystyle N}$ such that every even integer ${\displaystyle n}$ larger than ${\displaystyle N}$ is a sum of a prime less than or equal to ${\displaystyle n^{0.95}}$ and a number with at most two prime factors.

In 2022, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:

Every even number greater than ${\displaystyle e^{e^{32.7}}\approx 1.4\cdot {10}^{69057979807814}}$ can be represented as the sum of a prime and a square-free number with at most two prime factors.

which refined upon an earlier result by Tomohiro Yamada. Also in 2024, Bordignon and Starichkova showed that the bound can be lowered to ${\displaystyle e^{e^{15.85}}\approx 3.6\cdot {10}^{3321634}}$ assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions.

In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer ${\displaystyle N}$ can be represented as

${\displaystyle N=p+2a,}$

where ${\displaystyle p}$ is prime and ${\displaystyle a}$ has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing ${\displaystyle N-p}$ to be even. Li's result can be viewed as an approximation to Lemoine's conjecture.