Chen's theorem

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.

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:^[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 n^0.95 and a number with at most two prime factors.

Notes

^ 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.
^ ^a ^b 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.

References

Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 0-387-94656-X. Chapter 10.
Wang, Yuan (1984). Goldbach conjecture. World Scientific. ISBN 9971-966-09-3.

External links

Jean-Claude Evard, Almost twin primes and Chen's theorem
Weisstein, Eric W. "Chen's Theorem". MathWorld.

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[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.

[Chen_1973-2] 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. (2). 10, 4 (4): 500–506. doi:10.1112/jlms/s2-10.4.500.

[4] Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18 (3): 597–604. doi:10.1007/s101140200168.

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