Editing Chen's theorem

{{Short description|Every large even number is either sum of a prime and a semi-prime or two primes}}
[[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]], 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 [[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=11 | issue=9 | year=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 in 1975.<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. |series=Series 2 | volume=10,4 | year=1975 | pages=500–506 | doi=10.1112/jlms/s2-10.4.500 | issue=4}}</ref> Chen's theorem is a significant step towards [[Goldbach's  conjecture]], and a celebrated application of [[sieve theory|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.<ref>[http://www-groups.dcs.st-and.ac.uk/history/Biographies/Renyi.html University of St Andrews - Alfréd Rényi]</ref><ref name="Alfréd Rényi 1948">{{cite journal | last=Rényi | first=A. A. | language=Russian | title=On the representation of an even number as the sum of a prime and an almost prime | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | volume=12 | pages=57–78 | year=1948 }}</ref>

== Variations ==
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''&thinsp;+&thinsp;''h'' is either prime or the product of two primes.

Ying Chun 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 | year=2002 | pages=597–604 | doi=10.1007/s101140200168 | issue=3| s2cid=121177443 }}</ref>

{{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 2025, 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.''}}
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 [[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>

: <math> N=p+2a, </math>

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 ==
=== Citations ===
{{Reflist|2}}

=== Books ===
* {{cite book | title = Additive Number Theory: the Classical Bases | volume = 164 | series = [[Graduate Texts in Mathematics]] | first = Melvyn B. | last = Nathanson | publisher = Springer-Verlag | year = 1996 | isbn=0-387-94656-X }} Chapter 10.
* {{cite book | last = Wang | first = Yuan | title = Goldbach conjecture | publisher = [[World Scientific]] | year=1984 | isbn = 9971-966-09-3 }}

== External links ==
* Jean-Claude Evard, [https://web.archive.org/web/20050901105944/http://www.math.utoledo.edu/~jevard/Page015.htm Almost twin primes and Chen's theorem]
* {{MathWorld |urlname = ChensTheorem |title = Chen's Theorem}}

[[Category:Theorems in analytic number theory]]
[[Category:Theorems about prime numbers]]
[[Category:Chinese mathematical discoveries]]