An Empirical Verification of Waring’s Problem for Fourth Powers: Confirming g(4) = 19 and G(4) = 16

Abstract

Waring’s Problem, a cornerstone of additive number theory, concerns the rep- resentation of integers as sums of k-th powers. For fourth powers (k = 4), it is classically known that g(4) = 19 and G(4) = 16, meaning every positive integer is a sum of 19 fourth powers, but only a finite set of integers require more than 16 terms. This paper presents a computational verification of these results through the implementation of a dynamic programming algorithm that determines the minimal number of fourth powers required to represent each integer up to a bound of 107. The computations successfully reproduces the complete, finite list of integers that require 17, 18, and 19 fourth powers, thereby empirically confirming the values of g(4) and the threshold for G(4). Furthermore, a statistical analysis of large integers demonstrates the efficacy of a greedy algorithm in finding representations with 16 terms, providing strong empirical support for the asymptotic result G(4) = 16. This work serves as a bridge between classical number theory and experimental mathe- matics, offering a reproducible and illustrative verification of profound theoretical truths.