On the representation of large even integers as the sum of eight primes from positive density sets

TL;DR

This paper proves that large even integers can be expressed as the sum of eight primes from a subset of primes with density greater than 1/2, establishing a sharp density threshold for such representations.

Contribution

It establishes a sharp density threshold of 1/2 for subsets of primes to represent large even integers as sums of eight primes from the subset.

Findings

Sets of primes with density > 1/2 can represent all large even integers as sums of eight primes.

The density threshold of 1/2 is optimal for such representations.

The result extends understanding of additive properties of dense prime subsets.

Abstract

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 1/2, then every sufficiently large even integer can be written as the sum of eight primes from A. The constant 1/2 in this statement is the best possible.