Chen's theorem explained

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. 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 implies there are gaps in Yamada's proof, which Bordignon overcomes in his PhD. thesis.^[8]

References

Books

• Book: Nathanson, Melvyn B. . Additive Number Theory: the Classical Bases . 164 . . Springer-Verlag . 1996 . 0-387-94656-X . Chapter 10.
• Book: Wang, Yuan . Goldbach conjecture . . 1984 . 9971-966-09-3 .

External links

• Jean-Claude Evard, Almost twin primes and Chen's theorem

Notes and References

1. Chen . J.R. . 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 . 1966 . 385–386.
2. Chen . J.R. . 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 . 1973 . 157–176.
3. Ross . P.M. . 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 . 1975 . 500–506 . 10.1112/jlms/s2-10.4.500 . 4.
4. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Renyi.html University of St Andrews - Alfréd Rényi
5. Rényi . A. A. . Russian . On the representation of an even number as the sum of a prime and an almost prime . Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya . 12 . 57–78 . 1948 .
6. Cai . Y.C. . Chen's Theorem with Small Primes. Acta Mathematica Sinica . 18 . 2002 . 597–604 . 10.1007/s101140200168 . 3. 121177443 .
7. Yamada . Tomohiro . 1511.03409 . Explicit Chen's theorem . math.NT . 2015-11-11.
8. Bordignon . Matteo . An Explicit Version of Chen's Theorem . Bulletin of the Australian Mathematical Society . Cambridge University Press (CUP) . 105 . 2 . 2022-02-08 . 0004-9727 . 10.1017/s0004972721001301 . 344–346. free . 2207.09452 .