Chen's theorem

The set of Chen - named after the Chinese mathematician Chen Jingrun - is a sentence from the number theory . It is usually stated as follows:

Any sufficiently large even number can be written as the sum of a prime number and a number with at most two prime factors.

It is considered the best approximation to date to a proof of the as yet unproven Goldbach Hypothesis , which states that every even number is the sum of two prime numbers.

background

See also the article on Goldbach's Hypothesis

Goldbach's conjecture is still unproven today. In the twentieth century, however, the first evidence of "similar" statements came. These say, for example, that every even number, or a certain subset of the even numbers, can be written as a sum of at most X prime numbers or of numbers with at most X prime factors.

In this sense, Chen Jingrun achieved the “best” approximation to the actual Goldbach conjecture in 1966 by proving the above-mentioned theorem.

The addition “sufficiently large” means that the rate applies to all even numbers above a certain minimum number.

content

The sentence in its original formulation deals with the question of how many different ways the even number can be represented as a corresponding sum. For this number he delivers the following minimum amount: ${\displaystyle x}$ ${\displaystyle P_{x}}$

${\displaystyle P_{x}\geq {\frac {0.67xC_{x}}{(\log x)^{2}}}}$

With

${\displaystyle C_{x}=\prod _{p>2}\left(1-{\frac {1}{{(p-1)}^{2}}}\right)\prod _{2<p\mid x}{\frac {p-1}{p-2}}}$

Summary of the evidence.

The English translation from 1973 contains another sentence (with proof) from the environment of the twin prime conjecture : for every difference (for which the twin prime conjecture is ) there is an infinite number of prime numbers for which there is a prime number or a product of two prime numbers. ${\displaystyle h}$ ${\displaystyle h=2}$ ${\displaystyle p}$ ${\displaystyle p+h}$

Further developments

In 1975 P. Ross published a simpler proof of Chen's theorem.

In 2002 YC Cai proved that one can represent (at least above a further limit) any even number in such a way that the summand, which is the prime number, is less than . ${\displaystyle n}$ ${\displaystyle n^{0,95}}$

Individual evidence

1. ↑ On the representation of a large even integer as the sum of a prime and a product of at most two primes. In: Kexue Tongbao. Volume 17, 1966, pp. 385-386 (chin.)
2. ↑ On the representation of a large even integer as the sum of a prime and a product of at most two primes. In: Scientia Sinica . Volume 16, 1973, pp. 157-176.
3. ^ A summary of the proof of Chen's theorem , Eugene Eisenstein, Lalit Jain, Adam Felix, 2004
4. ^ Ross, PM (1975). "On Chen's theorem that each large even number has the form (p1 + p2) or (p1 + p2p3)". J. London Math. Soc. (2) 10.4 : 500-506. doi: 10.1112 / jlms / s2-10.4.500
5. ^ YC Cai: Chen's Theorem with Small Primes. In: Acta Mathematica Sinica. 2000, Volume 18, Pages 597-604 ( doi: 10.1007 / s101140200168 ).