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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). 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.[3] Chen's theorem is a giant step towards the Goldbach conjecture, and a remarkable result of the sieve methods. Cai proved the following in 2002:[4] 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. 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. Notes 1. ^ 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 17: 385–386.
* Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. 164. SpringerVerlag. ISBN 038794656X. Chapter 10.
* JeanClaude Evard, Almost twin primes and Chen's theorem
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