On Waring's problem in sums of three cubes for smaller powers

TL;DR

This paper establishes an upper bound on the minimum number of positive k-th powers needed to represent large integers as sums of three positive cubes, for powers 2 to 4, advancing understanding of Waring's problem.

Contribution

It provides new upper bounds for the minimal s in Waring's problem when representing integers as sums of k-th powers of sums of three positive cubes for 2 ≤ k ≤ 4.

Findings

Derived upper bounds for s in the specified cases.

Extended Waring's problem to sums of three cubes for smaller powers.

Improved understanding of representations involving sums of three cubes.

Abstract

We give an upper bound for the minimum $s$ with the property that every sufficiently large integer can be represented as the sum of $s$ positive $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$