An update on the Linnik--Goldbach and Romanov problems

TL;DR

This paper advances the understanding of the Linnik--Goldbach problem by reducing the number of powers of two needed under GRH and improves bounds on Romanov's constant unconditionally.

Contribution

It demonstrates that under GRH, six powers of two suffice for representing large even integers, and it refines the bounds on Romanov's constant unconditionally.

Findings

Under GRH, six powers of two are sufficient for the Linnik--Goldbach problem.

Unconditionally, more than 25% of odd numbers can be expressed as a prime plus a power of two.

Updated bounds on Romanov's constant are provided.

Abstract

We consider the Linnik--Goldbach problem of writing all large even integers as the sum of two primes and a fixed number of powers of 2. We show that, under the generalised Riemann hypothesis, one can use 6 powers of two. In addition, we update the best known bounds on Romanov's constant, showing unconditionally that more than $25\%$ of odd numbers can be written as the sum of a prime and a power of 2.