Expression for $g(k)$ Related to Waring's Problem

TL;DR

This paper proves that a key condition related to Waring's problem never occurs, thereby establishing a fixed value for the function g(k) for all positive integers k.

Contribution

The paper provides a proof that a specific condition in Waring's problem does not occur, enabling an explicit determination of g(k).

Findings

Confirmed the non-occurrence of the key condition in Waring's problem.

Established a fixed value for g(k) for all positive integers k.

Simplified the understanding of the function g(k) in Waring's problem.

Abstract

Waring's Problem asks whether, for each positive integer $k$, there exists an integer $s$ such that every positive integer is a sum of at most $k$th powers. While Hilbert proved the existence of such $s$, Waring's Problem has lead to areas of related work, namely the function $g(k)$, which denotes the least such $s$. There is no known general closed form for $g(k)$, though for $g(k)$ has been evaluated for small $k$. Prior work has reduced the problem to verifying a particular condition, which if never occurs, implies an expression for $g(k)$. In this paper, I present a proof the condition never occurs, thus fixing the value of $g(k)$.