“Chen's theorem”的意思、由来-开放百科全书

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.

Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had showed 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]

Variations

Chen's 1973 paper stated two results with nearly identical proofs.^[2]{{Rp|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.

Tomohiro Yamada proved the following explicit version of Chen's theorem in 2015:^[6]

References

Citations

1. ^{{cite journal | last=Chen | first=J.R. | title=On the representation of a large even integer as the sum of a prime and the product of at most two primes | journal=Kexue Tongbao | volume=11 | issue=9 | year=1966 | pages=385–386}}
2. ^¹{{cite journal | last=Chen | first=J.R. | title=On the representation of a larger even integer as the sum of a prime and the product of at most two primes | journal=Sci. Sinica | volume=16 | year=1973 | pages=157–176}}
3. ^{{cite journal | last=Ross | first=P.M. | title=On Chen's theorem that each large even number has the form (p₁+p₂) or (p₁+p₂p₃) | journal=J. London Math. Soc. |series=Series 2 | volume=10,4 | year=1975 | pages=500–506 | doi=10.1112/jlms/s2-10.4.500 | issue=4}}
4. ^University of St Andrews - Alfréd Rényi
5. ^{{cite journal | last=Cai | first=Y.C. | title=Chen's Theorem with Small Primes| journal=Acta Mathematica Sinica | volume=18 | year=2002 | pages=597–604 | doi=10.1007/s101140200168 | issue=3}}
6. ^{{cite arXiv|last=Yamada |first=Tomohiro |eprint=1511.03409 |title=Explicit Chen's theorem |class=math.NT |date=2015-11-11}}

History

Variations

References

Citations

Books

External links