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.
We will present the proof of Chen’s Theorem with a “top-down” approach: after having analyzed the statement in detail, we will analyze the main techniques used by Chen by following an example, until we’ll break down the initial problem into three subproblems that will be solved separately, as Chen himself did. We will introduce some important number theory notions and theorems as needed. The required prerequisites are, as in all of our site, a good knowledge of school mathematics and the basics of real analysis in one variable (derivatives, integrals, continuity, etc.). Other prerequisites will be indicated at the beginning of each article; for example the first article, like all the rest of the path, assumes a certain knowledge of asymptotic analysis, that we have covered on our site in the path Complementary material.