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 giant step towards the Goldbach's conjecture, and a remarkable result of the 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]

Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015:^[7]

Every even number greater than ${\displaystyle e^{e^{36}}\approx 1.7\cdot 10^{1872344071119343}}$ is the sum of a prime and a product of at most two primes.

In 2022, Matteo Bordignon implies there are gaps in Yamada's proof, which Bordignon overcomes in his PhD. thesis.^[8]