On Waring's problem for larger powers

TL;DR

This paper establishes an improved upper bound for the minimal number of k-th powers needed to represent sufficiently large numbers, advancing understanding of Waring's problem for larger powers.

Contribution

The authors introduce new methods that improve existing bounds on Waring's problem for all k ≥ 14, providing tighter estimates for G(k).

Findings

Established that G(k) ≤ ⌈k(log k + 4.20032)⌉ for all k.

Improved bounds for G(k) when k ≥ 14.

Enhanced methods for analyzing sums of k-th powers.

Abstract

Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has \[ G(k)\le \lceil k(\log k+4.20032)\rceil . \] Our new methods improve on all bounds available hitherto when $k\ge 14$.