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number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, Chen's theorem states that every sufficiently large
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
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 ...
(the product of two primes).


History

The theorem 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, structure, space, models, and change. History On ...
Chen Jingrun Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son i ...
in 1966, with further details of the proof in 1973. His original proof was much simplified by P. M. Ross in 1975. Chen's theorem is a giant step towards the
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hol ...
, and a remarkable result of the 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. It states that if ''h'' is a positive even
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, 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: Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015: Matteo Bordignon implies there are gaps in Yamada's proof, which Bordignon overcomes in his PhD. thesis.


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