A density version of Waring-Goldbach problem

TL;DR

This paper investigates a density-based variant of the Waring-Goldbach problem, demonstrating that large numbers meeting certain conditions can be expressed as sums of prime k-th powers from a dense subset of primes.

Contribution

It introduces a density condition on subsets of primes and proves that such subsets can represent large numbers as sums of prime k-th powers, extending classical results.

Findings

Sets of primes with density > 1 - 1/2k can represent large numbers as sums of prime k-th powers.

The result applies to sufficiently large natural numbers satisfying necessary congruence conditions.

The paper generalizes the Waring-Goldbach problem to dense subsets of primes.

Abstract

In this paper, we study a density version of the Waring-Goldbach problem. Suppose that A is a subset of the primes, and the lower density of A in the primes is larger than 1-1/2k. We prove that every sufficiently large natural number n satisfying the necessary congruence condition can be expressed as a sum of s terms of the k-th powers of primes from set A, where s is a positive integer dependent on k.