## The goal

The goal of this path is to prove one of the theorems most similar to Goldbach’s conjecture, Chen’s Theorem, the statement of which can be expressed as follows:

*Every sufficiently large even number is the sum of either two primes, or a prime and a semiprime (that is, a product of two primes).*

This Theorem was proved for the first time by the Chinese mathematician Chen Jingrun in 1966 (subsequently the proof was refined both by himself and by other mathematicians). The proof is based on a mathematical theory called sieve theory, developed a few years earlier but with very ancient origins, which can be traced back to the mathematician of ancient Greece Eratosthenes of Cyrene, inventor of the homonymous sieve, the first ever. Clearly, since then sieve theory has evolved a lot, but it has always remained an “elementary” theory, i.e. one not based on complex analysis.

## The path

The proof of Chen’s Theorem combines various sieve theory techniques, which will be introduced and explored separately starting from the second article, until a good part of the path. Subsequently we’ll go into the details of the proof, guided by an example, until we’ll break down the initial problem into three subproblems that will be solved separately. Occasionally we’ll refer to theorems or concepts of number theory, for which from time to time in-depth links will be proposed.

General introduction to sieve theory paths