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Integers that are not the sum of positive powers

Brennan Benfield, Oliver Lippard

TL;DR

This work advances the generalized Waring problem by combining computational sieving with partition-theoretic methods to characterize the sets of integers not expressible as sums of $j$ positive $k$-th powers. It unifies Zenkin’s framework with new $n^*$-based criteria to propagate representations to larger numbers and computes the obstruction sets $\mathbf{B}^k_j$ for $3\le j\le9$ across several $k$, proving Zenkin’s conjecture $g(1,k)=57$ and deriving both unconditional and conjectural bounds for $G(1,k)$. The paper presents detailed proofs for key cases (notably $k=5$–$9$) and develops general properties of $\mathbf{B}$-sets, including chain-inclusion and stabilization phenomena, supported by extensive computational data. The results refine our understanding of when every sufficiently large integer is representable as a sum of $j$ $k$-th powers and illuminate the structure and growth of nonrepresentable sets, with implications for future bounds in Waring-type problems. The methods blend algorithmic number theory with analytic and combinatorial tools to push the boundaries of known $G(1,k)$ and $g(1,k)$ values.

Abstract

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 $5 \le k \le 9$, 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.

Integers that are not the sum of positive powers

TL;DR

This work advances the generalized Waring problem by combining computational sieving with partition-theoretic methods to characterize the sets of integers not expressible as sums of positive -th powers. It unifies Zenkin’s framework with new -based criteria to propagate representations to larger numbers and computes the obstruction sets for across several , proving Zenkin’s conjecture and deriving both unconditional and conjectural bounds for . The paper presents detailed proofs for key cases (notably –) and develops general properties of -sets, including chain-inclusion and stabilization phenomena, supported by extensive computational data. The results refine our understanding of when every sufficiently large integer is representable as a sum of -th powers and illuminate the structure and growth of nonrepresentable sets, with implications for future bounds in Waring-type problems. The methods blend algorithmic number theory with analytic and combinatorial tools to push the boundaries of known and values.

Abstract

The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of positive -th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases , 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 positive -th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where . 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.
Paper Structure (20 sections, 33 theorems, 47 equations, 3 tables)

This paper contains 20 sections, 33 theorems, 47 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Computational Methods
  3. Existence of Representations
  4. Computing $n^*$
  5. The Known Cases: Squares, Cubes, & Fourth Powers
  6. Squares
  7. Cubes
  8. Fourth Powers
  9. Positive Fifth Powers
  10. Positive Sixth Powers
  11. Positive Seventh Powers
  12. Positive Eighth Powers
  13. Positive Ninth Powers
  14. General Properties of B-sets
  15. The size of the set $\mathbf{B}^k$ as $k$ gets large
  16. ...and 5 more sections

Key Result

Theorem 1.1

(Zenkin Zenkin3) For every $k \geq 2$, there exists an index $G(1,k)$ such that for all $j \geq G(1,k)$, all but finitely many positive integers are $(j,k)$-representable.

Theorems & Definitions (68)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 4.1
  • Lemma 4.2: Deshouillers, Hennecart, Kawada, Landreau, Wooley
  • Theorem 4.3: Zenkin Zenkin3
  • ...and 58 more