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'''Chen's theorem''' states that every sufficiently large even number can be written as the sum of either two [[prime number|primes]], or a prime and a [[semiprime]] (the product of two primes). The [[theorem]] was first stated by [[China|Chinese]] [[mathematician]] [[Chen Jingrun]] in 1966,<ref>{{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=17 | date=1966 | pages=385–386}}</ref> with further details of the [[mathematical proof|proof]] in 1973.<ref name="Chen 1973">{{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}}</ref> His original proof was much simplified by P. M. Ross.<ref>{{cite journal | last=Ross | first=P.M. | title=On Chen's theorem that each large even number has the form (p<sub>1</sub>+p<sub>2</sub>) or (p<sub>1</sub>+p<sub>2</sub>p<sub>3</sub>) | journal=J. London Math. Soc. (2) | volume=10,4 | year=1975 | pages=500--506 | doi=10.1112/jlms/s2-10.4.500}}</ref> Chen's theorem is a giant step towards the [[Goldbach conjecture]], and a remarkable result of the [[sieve methods]]. |
'''Chen's theorem''' states that every sufficiently large even number can be written as the sum of either two [[prime number|primes]], or a prime and a [[semiprime]] (the product of two primes). The [[theorem]] was first stated by [[China|Chinese]] [[mathematician]] [[Chen Jingrun]] in 1966,<ref>{{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=17 | date=1966 | pages=385–386}}</ref> with further details of the [[mathematical proof|proof]] in 1973.<ref name="Chen 1973">{{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}}</ref> His original proof was much simplified by P. M. Ross.<ref>{{cite journal | last=Ross | first=P.M. | title=On Chen's theorem that each large even number has the form (p<sub>1</sub>+p<sub>2</sub>) or (p<sub>1</sub>+p<sub>2</sub>p<sub>3</sub>) | journal=J. London Math. Soc. (2) | volume=10,4 | year=1975 | pages=500--506 | doi=10.1112/jlms/s2-10.4.500}}</ref> Chen's theorem is a giant step towards the [[Goldbach conjecture]], and a remarkable result of the [[sieve methods]]. |
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Cai proved the following in 2002.<ref>{{cite journal | last=Cai | first=Y.C. | title=Chen’s Theorem with Small Primes| journal=Acta Mathematica Sinica | volume=18 | date=2002 | pages=597–604}}</ref> |
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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<sup>0.95'' |
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</sup> and a number with at most two prime factors. |
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Chen's 1973 paper stated two results with nearly identical proofs.<ref name="Chen 1973" />{{Rp|158}} His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the [[Twin prime|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. |
Chen's 1973 paper stated two results with nearly identical proofs.<ref name="Chen 1973" />{{Rp|158}} His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the [[Twin prime|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. |
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Revision as of 14:37, 1 February 2010
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.
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 N0.95 and a number with at most two prime factors.
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.
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. 17: 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: 500--506. doi:10.1112/jlms/s2-10.4.500.
- ^ Cai, Y.C. (2002). "Chen's Theorem with Small Primes". Acta Mathematica Sinica. 18: 597–604.
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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