Chen's theorem
Encyclopedia
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). The theorem
was first stated by Chinese
mathematician
Chen Jingrun
in 1966, with further details of the proof
in 1973. His original proof was much simplified by P. M. Ross. Chen's theorem is a giant step towards the Goldbach conjecture, and a remarkable result of the sieve methods.
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, 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:
There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n0.95 and a number with at most two prime factors.
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, or a prime and a semiprime
Semiprime
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....
(the product of two primes). The theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
was first stated by Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Chen Jingrun
Chen Jingrun
Chen Jingrun was a Chinese mathematician who made significant contributions to number theory.- Personal life :Chen was the third son in a large family from Fuzhou, Fujian, China. His father was a postal worker. Chen Jingrun graduated from the Mathematics Department of Xiamen University in 1953...
in 1966, with further details of the proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
in 1973. His original proof was much simplified by P. M. Ross. Chen's theorem is a giant step towards the Goldbach conjecture, and a remarkable result of the sieve methods.
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
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
. 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:
There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n0.95 and a number with at most two prime factors.
External links
- Jean-Claude Evard, Almost twin primes and Chen's theorem