The Exceptional Set in Goldbach's Problem with two Chen Primes

TL;DR

This paper proves that all integers congruent to 4 mod 6 can be expressed as the sum of two Chen primes, with only a small set of exceptions, using advanced sieving techniques and probabilistic models.

Contribution

It introduces a novel sieving approach and probabilistic modeling to improve understanding of sums involving Chen primes, approaching optimality under current conjectures.

Findings

All numbers n ≡ 4 mod 6 are sums of two Chen primes except for a small set.

Developed an efficient sieving strategy using a power-saving Bombieri--Vinogradov variant.

Showed primes can be approximated in additive problems by the Cramér model with a power-sized sifting parameter.

Abstract

We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is optimal, barring substantial progress on the twin prime or binary Goldbach conjectures. The proof is based on constructing a non-negative model for the Chen primes in a suitable approximate sense. To do this, we develop an efficient sieving strategy that makes use of a power-saving variant of the Bombieri--Vinogradov theorem. Furthermore, we show that the primes are well approximated in additive problems by the Cram\'er model (rough numbers) with a sifting parameter of power size.