The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original proof was much simplified by P. M. Ross. Chen's theorem is a giant step towards the Goldbach conjecture, and a remarkable result of the sieve methods.
Chen's 1973 paper stated two results with nearly identical proofs.: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:
- There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n0.95 and a number with at most two prime factors.
Tomohiro Yamada proved the following explicit version of Chen's theorem in 2015:
- Every even number greater than is the sum of a prime and a product of at most two primes.
- Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao 11 (9): 385–386.
- Chen, J.R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica 16: 157–176.
- Ross, P.M. (1975). "On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3)". J. London Math. Soc. (2) 10,4 (4): 500–506. doi:10.1112/jlms/s2-10.4.500.
- Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica 18 (3): 597–604. doi:10.1007/s101140200168.
- Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].
- Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics 164. Springer-Verlag. ISBN 0-387-94656-X. Chapter 10.
- Wang, Yuan (1984). Goldbach conjecture. World Scientific. ISBN 9971-966-09-3.