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).

The statue of Chen Jingrun at Xiamen University.

History

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 showed 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]

Variations

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:[5]

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:[6]

Every even number greater than is the sum of a prime and a product of at most two primes.

References

Citations
  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. 11 (9): 385–386.
  2. 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.
  3. 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. Series 2. 10, 4 (4): 500–506. doi:10.1112/jlms/s2-10.4.500.
  4. University of St Andrews - Alfréd Rényi
  5. Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18 (3): 597–604. doi:10.1007/s101140200168.
  6. Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].

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