General closed form for g(k) in Waring’s problem

Determine a general closed form expression for g(k), where g(k) denotes the least integer s such that every positive integer can be expressed as a sum of at most s k-th powers of positive integers.

Background

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 s k-th powers. Hilbert proved the existence of such an s, and g(k) denotes the minimal such s. While values of g(k) have been determined for small k, a general closed form for g(k) has historically eluded researchers.

The paper discusses a specific candidate formula for g(k) and argues that a certain obstructing condition never occurs, thereby claiming an exact expression. Prior to such claims, the absence of a general closed form was a central unresolved question in the area.