Chen's theorem: Difference between revisions
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[[File:Chen Jing-run.JPG|right|thumb|The statue of Chen Jingrun at [[Xiamen University]].]] | [[File:Chen Jing-run.JPG|right|thumb|The statue of Chen Jingrun at [[Xiamen University]].]] | ||
In [[number theory]], '''Chen's theorem''' states that every sufficiently large [[parity (mathematics)|even]] number can be written as the sum of either two [[prime number|primes]] | In [[number theory]], '''Chen's theorem''' states that every sufficiently large [[parity (mathematics)|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). | ||
It is a weakened form of [[Goldbach's conjecture]], which states that every even number is the sum of two primes. | It is a weakened form of [[Goldbach's conjecture]], which states that every even number is the sum of two primes. | ||
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{{bi|left=1.6|''There exists a natural number <math>N</math> such that every even integer <math>n</math> larger than <math>N</math> is a sum of a prime less than or equal to <math>n^{0.95}</math> and a number with at most two prime factors.''}} | {{bi|left=1.6|''There exists a natural number <math>N</math> such that every even integer <math>n</math> larger than <math>N</math> is a sum of a prime less than or equal to <math>n^{0.95}</math> and a number with at most two prime factors.''}} | ||
In | In 2022, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:<ref>{{cite arXiv|first1=Daniel R.|last1=Johnston|first2=Matteo|last2=Bordignon|first3=Valeriia|last3=Starichkova|eprint=2207.09452 |title=An explicit version of Chen's theorem |class=math.NT |date=2025-01-28}}</ref> | ||
{{bi|left=1.6|''Every even number greater than <math>e^{e^{32.7}} \approx 1.4 \cdot 10^{69057979807814}</math> can be represented as the sum of a prime and a square-free number with at most two prime factors.''}} | {{bi|left=1.6|''Every even number greater than <math>e^{e^{32.7}} \approx 1.4 \cdot 10^{69057979807814}</math> can be represented as the sum of a prime and a square-free number with at most two prime factors.''}} | ||
which refined upon an earlier result by Tomohiro Yamada.<ref>{{cite arXiv|last=Yamada |first=Tomohiro |eprint=1511.03409 |title=Explicit Chen's theorem |class=math.NT |date=2015-11-11}}</ref> | which refined upon an earlier result by Tomohiro Yamada.<ref>{{cite arXiv|last=Yamada |first=Tomohiro |eprint=1511.03409 |title=Explicit Chen's theorem |class=math.NT |date=2015-11-11}}</ref> | ||
Also in 2024, Bordignon and Starichkova<ref>{{Cite journal |first1=Matteo |last1=Bordignon |first2=Valeriia |last2=Starichkova |title=An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis|date=2024 | journal=The Ramanujan Journal | doi=10.1007/s11139-024-00866-x | volume=64 | pages=1213–1242|arxiv=2211.08844 }}</ref> showed that the bound can be lowered to <math>e^{e^{14}} \approx 2.5\cdot10^{522284}</math> assuming the [[ | Also in 2024, Bordignon and Starichkova<ref>{{Cite journal |first1=Matteo |last1=Bordignon |first2=Valeriia |last2=Starichkova |title=An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis|date=2024 | journal=The Ramanujan Journal | doi=10.1007/s11139-024-00866-x | volume=64 | pages=1213–1242|arxiv=2211.08844 }}</ref> showed that the bound can be lowered to <math>e^{e^{14}} \approx 2.5\cdot10^{522284}</math> assuming the [[generalized Riemann hypothesis]] (GRH) for [[Dirichlet L-function]]s. | ||
In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer <math>N</math> can be represented as<ref>{{cite journal | last=Li | first=H. | title=On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors| journal=Ramanujan J. | volume=49 | year=2019 | pages=141–158 }}</ref> | In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer <math>N</math> can be represented as<ref>{{cite journal | last=Li | first=H. | title=On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors| journal=Ramanujan J. | volume=49 | year=2019 | pages=141–158 }}</ref> | ||
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where <math>p</math> is prime and <math>a</math> has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing <math> N-p </math> to be even. Li's result can be viewed as an approximation to [[Lemoine's conjecture]]. | where <math>p</math> is prime and <math>a</math> has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing <math> N-p </math> to be even. Li's result can be viewed as an approximation to [[Lemoine's conjecture]]. | ||
== References == | == References == | ||
=== Citations === | === Citations === | ||
Latest revision as of 12:19, 26 June 2025
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).
It is a weakened form of Goldbach's conjecture, which states that every even number is the sum 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 significant step towards Goldbach's conjecture, and a celebrated application of sieve methods.
Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown 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][5]
Variations
Chen's 1973 paper stated two results with nearly identical proofs.[2]Template:Rp 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:[6]
In 2022, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:[7] Template:Bi which refined upon an earlier result by Tomohiro Yamada.[8] Also in 2024, Bordignon and Starichkova[9] showed that the bound can be lowered to assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions.
In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer can be represented as[10]
where is prime and has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing to be even. Li's result can be viewed as an approximation to Lemoine's conjecture.
References
Citations
Books
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External links
- Jean-Claude Evard, Almost twin primes and Chen's theorem
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- ↑ University of St Andrews - Alfréd Rényi
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