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.
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.[4][5]
Variations
Chen's 1973 paper stated two results with nearly identical proofs.[2]Шаблон: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.
Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015:[7]Шаблон:Bi \approx 1.7\cdot10^{1872344071119343}</math> is the sum of a prime and a product of at most two primes.}} In 2022, Matteo Bordignon implies there are gaps in Yamada's proof, which Bordignon overcomes in his PhD. thesis.[8]
