An explicit version of Chen's theorem and the linear sieve

TL;DR

This paper proves that every sufficiently large even number can be expressed as the sum of a prime and a number with at most two prime factors, extending Chen's theorem with an explicit bound.

Contribution

It provides an explicit version of Chen's theorem, establishing a concrete lower bound for the even integers that can be represented in this form.

Findings

Every even integer larger than exp(exp(32.7)) can be written as a sum of a prime and a product of at most two primes.

The result gives an explicit bound improving previous non-quantitative versions.

The approach is inspired by Nathanson and Yamada's work on the linear sieve.

Abstract

Drawing inspiration from the work of Nathanson and Yamada we prove that every even integer larger than $\exp (\exp (32.7))$ can be written as the sum of a prime and the product of at most two primes.