Chen's theorem
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.
HistoryEdit
The theorem was first stated by Chinese mathematician Chen Jingrun in 1966,<ref>Template:Cite journal</ref> with further details of the proof in 1973.<ref name="Chen 1973">Template:Cite journal</ref> His original proof was much simplified by P. M. Ross in 1975.<ref>Template:Cite journal</ref> 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.<ref>University of St Andrews - Alfréd Rényi</ref><ref name="Alfréd Rényi 1948">Template:Cite journal</ref>
VariationsEdit
Chen's 1973 paper stated two results with nearly identical proofs.<ref name="Chen 1973" />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:<ref>Template:Cite journal</ref>
In 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:<ref>Template:Cite arXiv</ref> Template:Bi \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>Template:Cite arXiv</ref> Also in 2024, Bordignon and Starichkova<ref>Template:Cite journal</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-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 <math>N</math> can be represented as<ref>Template:Cite journal</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.
ReferencesEdit
CitationsEdit
BooksEdit
- Template:Cite book Chapter 10.
- Template:Cite book
External linksEdit
- Jean-Claude Evard, Almost twin primes and Chen's theorem
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