2024/04/12 by Brennan Benfield, Benfield, Brennan, Oliver Lippard +1
Engineering · Mathematics · #11-04 #11P05 #11P81 #11Y50 #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2404.08193
openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of j positive k-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases 5 ≤ k ≤ 9, and resolves conjectures of Zenkin and the OEIS. This paper further establishes theoretical results regarding the properties of the sets of integers that are not the sum of j positive k-th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where G(1,k) < j < g(1,k). Improved computational techniques and results from Waring's problem are used throughout to catalog the sets of such integers, which are then considered in a general setting.