A density version of quaternary Goldbach problem

TL;DR

This paper proves that for sufficiently large even integers, four primes from specified subsets with certain density conditions can sum to the integer, extending Goldbach-type results.

Contribution

It establishes a density condition under which four primes from subsets can sum to large even integers, generalizing classical Goldbach problems.

Findings

The sum of densities exceeds 1 for pairs of subsets.

Existence of four primes summing to large even integers under density conditions.

The density condition is proven to be optimal.

Abstract

Let $\mathcal{P}$ denote the set of all primes, and let $\underline\delta(P)$ denote the relative lower density of a subset $P$ in $\mathcal{P}$. Suppose that $P_1, P_2, P_3, P_4$ are four subsets of primes with $\underline\delta(P_1)+\underline\delta(P_2)>1$ and $ \underline\delta(P_3)+\underline\delta(P_4)>1.$ Then for every sufficiently large even integer $n$, there exist primes $p_i \in P_i$ $(i=1,2,3,4)$ such that $n=p_1+p_2+p_3+p_4$. The condition is the best possible.