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Combinatorial designs and the Prouhet--Tarry--Escott problem

Munenori Inagaki, Hideki Matsumura, Masanori Sawa, Yukihiro Uchida

Abstract

This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE$_r$ solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE$_r$ solutions: one is a combinatorial composition that produces PTE$_r$ solutions by embedding lower-dimensional solutions into OAs with $r$ columns, and the other is a recursive technique in which a PET$_r$ solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE$_1$ and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory.

Combinatorial designs and the Prouhet--Tarry--Escott problem

Abstract

This is the first paper that provides a systematic treatment of the -dimensional PTE problem in additive number theory, abbreviated by PTE, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE solutions: one is a combinatorial composition that produces PTE solutions by embedding lower-dimensional solutions into OAs with columns, and the other is a recursive technique in which a PET solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory.
Paper Structure (17 sections, 24 theorems, 114 equations)

This paper contains 17 sections, 24 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $r$-dimensional PTE problem and combinatorially nontrivial solutions
  4. Combinatorial arrays and block designs
  5. Combinatorial array
  6. Block design
  7. Combinatorial PTE inequality and tight solutions
  8. Proof of the combinatorial PTE inequality
  9. Tight solutions associated with distinctive $t$-designs and OA
  10. Direct constructions of $\mathop{\mathrm{PTE}}\nolimits_r$ solutions and disjoint combinatorial designs
  11. OA-based direct construction
  12. Block-design-based direct construction
  13. Lifting constructions
  14. Combinatorial array lifting and Borwein solution
  15. Cartesian product lifting and Jacroux's partitioning
  16. ...and 2 more sections

Key Result

Theorem 2.5

There exists an ideal solution of the $\mathop{\mathrm{PTE}}\nolimits_2$ given by where

Theorems & Definitions (83)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4: PIL2023
  • Theorem 2.5: MS2025, Two-dimensional Borwein solution
  • Remark 2.6
  • Proposition 2.7: Ghiglione
  • Theorem 2.8: Lorentz--Alpers--Tijdeman (LAT) construction
  • Definition 2.9: Nontrivial solution
  • Example 2.10: Halving the trivial OA
  • Example 2.11: Doubling the Fano plane
  • ...and 73 more