The statue of Chen Jingrun at Xiamen University.

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.

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 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]

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:[6]

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

Every even number greater than 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]

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. 1 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. ↑ Rényi, A. A. (1948). "On the representation of an even number as the sum of a prime and an almost prime". Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya (in Russian). 12: 57–78.
  6. ↑ Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18 (3): 597–604. doi:10.1007/s101140200168. S2CID 121177443.
  7. ↑ Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].
  8. ↑ Bordignon, Matteo (2022-02-08). "An Explict Version of Chen's Theorem". Bulletin of the Australian Mathematical Society. Cambridge University Press (CUP). 105 (2): 344–346. arXiv:2207.09452. doi:10.1017/s0004972721001301. ISSN 0004-9727.

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