Linnik's approximation to Goldbach's conjecture, and other problems

TL;DR

This paper investigates the problem of representing large even numbers as sums of two primes and a limited number of powers of 2, proposing an approach that nearly improves existing bounds and also refines estimates in related Waring-Goldbach problems.

Contribution

It introduces an approach that nearly advances the bounds on the number of powers of 2 needed in Goldbach-type representations and improves estimates in related additive problems.

Findings

Approach nearly improves bounds on powers of 2 in Goldbach representations.

Refines estimates in Waring-Goldbach problems.

Provides insights into Linnik's approximation to Goldbach's conjecture.

Abstract

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the current bounds on $K$. Finally, we improve the estimates in other Waring--Goldbach problems.