number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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 In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime n ...

(the product of two primes). It is a weakened form of

Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...

, which states that every even number is the sum of two primes.

History

The

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

was first stated by Chinese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Chen Jingrun in 1966, with further details of the

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

in 1973. His original proof was much simplified by P. M. Ross in 1975. 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 Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest to A ...

, 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.

Variations

Chen's 1973 paper stated two results with nearly identical proofs. His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the

twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' ...

. It states that if ''h'' is a positive even

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, 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: In 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem: which refined upon an earlier result by Tomohiro Yamada. Also in 2024, Bordignon and Starichkova showed that the bound can be lowered to

e^ \approx 2.5\cdot10^

assuming the

Generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...

(GRH) for

Dirichlet L-function In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By anal ...

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

N

can be represented as :

N=p+2a,

where

p

is prime and

a

has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing

N-p

to be even. Li's result can be viewed as an approximation to Lemoine's conjecture.

References

Citations

Books

* Chapter 10. *

External links

* Jean-Claude Evard
Almost twin primes and Chen's theorem
* {{MathWorld , urlname = ChensTheorem , title = Chen's Theorem Theorems in analytic number theory Theorems about prime numbers Chinese mathematical discoveries