Chen's theorem

In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).

It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes.

The theorem was first stated by Chinese mathematician Chen Jingrun in 1966,^[1] with further details of the proof in 1973.^[2] His original proof was much simplified by P. M. Ross in 1975.^[3] Chen's theorem is a significant step towards Goldbach's conjecture, and a celebrated application of sieve methods.

Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most K primes.^[4][5]

Chen's 1973 paper stated two results with nearly identical proofs.^[2]: 158 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:^[6]

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 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:^[7]

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.^[8] Also in 2024, Bordignon and Starichkova^[9] showed that the bound can be lowered to ${\displaystyle e^{e^{14}}\approx 2.5\cdot 10^{522284}}$ 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^[10]

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

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.

• Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 0-387-94656-X. Chapter 10.
• Wang, Yuan (1984). Goldbach conjecture. World Scientific. ISBN 9971-966-09-3.

• Jean-Claude Evard, Almost twin primes and Chen's theorem
• Weisstein, Eric W. "Chen's Theorem". MathWorld.