On Waring's problem for intermediate powers

TL;DR

This paper establishes upper bounds for the minimal number of positive integral powers needed to represent large numbers for powers 7 through 16, advancing understanding of Waring's problem.

Contribution

The paper provides new upper bounds for G(k) for intermediate powers 7 to 16, improving previous results in Waring's problem.

Findings

G(7) ≤ 31

G(8) ≤ 39

G(9) ≤ 47

Abstract

Let $G(k)$ denote the least number $s$ such that every sufficiently large natural number is the sum of at most $s$ positive integral $k$th powers. We show that $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$, $G(12)\le 72$, $G(13)\le 81$, $G(14)\le 90$, $G(15)\le 99$, $G(16)\le 108$.