An update on the Linnik--Goldbach and Romanov problems
Number Theory
2026-05-19 v1
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 of odd numbers can be written as the sum of a prime and a power of 2.
Cite
@article{arxiv.2605.17825,
title = {An update on the Linnik--Goldbach and Romanov problems},
author = {Daniel R. Johnston and Tim Trudgian},
journal= {arXiv preprint arXiv:2605.17825},
year = {2026}
}
Comments
RHB at 50 proved that seven 2's would do.\\ Twenty-five years later, we show six in this review.\\ We wish him to his century when proofs of 5 are due!