Chen's theorem

Chen Jingrun

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).

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 conjecture, and a remarkable result of the sieve methods.

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:^[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.

Tomohiro Yamada proved the following explicit version of Chen's theorem in 2015:^[5]

Every even number greater than ${\displaystyle e^{e^{36}}\approx 1.7\cdot 10^{1872344071119348}}$ 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. (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.
5. Yamada, Tomohiro (2015-11-11). "Explicit Chen's theorem". arXiv:1511.03409 [math.NT].

Books

• Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. 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.