Chen's theorem

File:Chen Jing-run.JPG

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, with further details of the proof in 1973. His original proof was much simplified by P. M. Ross. 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. 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:

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:

Every even number greater than $e^{e^{36}} \approx 1.7\cdot10^{1872344071119348}$ is the sum of a prime and a product of at most two primes.

References

Books

• Chapter 10.

External links

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