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]
In 2022, Matteo Bordignon found multiple errors in Yamada's proof, and provided an alternative proof for a lower bound: [8]
Also in 2022, Bordignon and Valeriia Starichkova [9] showed that the bound can be lowered to assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions. In 2024, Bordignon and Starichkova [10] improved this result by lowering the bound to .
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A timeline of number theory.
