A Density Theorem for Higher Order Sums of Prime Numbers

TL;DR

This paper proves that large even integers can be expressed as sums of four primes from subsets of primes with density greater than half, extending previous results to higher sums and multiple subsets using elementary combinatorics.

Contribution

It extends prime sum theorems to higher orders and multiple subsets, introducing new elementary combinatorial lemmas for the proofs.

Findings

Large even integers are sums of four primes from dense prime subsets.

Results generalize previous three-prime sum theorems to higher sums.

Elementary combinatorial lemmas are key to the proofs.

Abstract

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.