An update on the Linnik--Goldbach and Romanov problems

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.