Problems related to Waring-Goldbach problem involving cubes of primes

TL;DR

This paper explores the Waring-Goldbach problem involving cubes of primes, examining representations of various integers as sums of four prime cubes, and raises open questions to deepen understanding of these additive problems.

Contribution

It investigates whether integers like primes, prime squares, prime cubes, or even-number cubes can be expressed as sums of four prime cubes, and proposes new problems in this area.

Findings

Some integers can be expressed as sums of four prime cubes.

Examples suggest most integer cubes are representable as sums of four cubes.

Open problems are proposed to advance understanding of the problem.

Abstract

In this note, we try to understand the recent development on the Waring-Goldbach problem involving cubes of primes. Especially, we want to determine whether integers that are either primes, squares of primes, cubes of primes, or a cube of an even number can be written as the sum of four cubes of primes. Meanwhile, we raise some problems that may deepen our understanding of the problem about the sum of four cubes of primes. Moreover, some examples suggest that almost all the cubes of integers can be written as the sum of cubes of four integers.