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An Extension of the Prouhet-Tarry-Escott Problem to Nonnegative, Noninteger Powers

David Treeby, Edward Wang

TL;DR

The paper addresses extending the Prouhet–Tarry–Escott problem to noninteger powers of consecutive integers by constructing a signed-sum mechanism based on a Thue–Morse weighted polynomial family. It establishes exact vanishing for integer powers via a differencing operator leading to $f^j_n(x)=0$ when $n>j$ (and demonstrates this explicitly for $j=0,1,2,3$ with $n=4$), and then extends the result to noninteger $j$ using Newton's generalized binomial theorem, showing the signed sums decay as $O(x^{j-n})$ as $x\to\infty$ and thus approach zero. The main contributions are the constructive Thue–Morse framework for integer powers, the noninteger generalization (Theorem thm3), and the asymptotic decay guarantee that enables near-zero signed sums for large bases. This advances the scope of PTE-type identities beyond integers and provides a method to approximate equalities for noninteger exponents in partition problems.

Abstract

We present an extension of the Prouhet-Tarry-Escott problem by demonstrating that signed sums of noninteger powers of consecutive integers can be made arbitrarily close to zero.

An Extension of the Prouhet-Tarry-Escott Problem to Nonnegative, Noninteger Powers

TL;DR

The paper addresses extending the Prouhet–Tarry–Escott problem to noninteger powers of consecutive integers by constructing a signed-sum mechanism based on a Thue–Morse weighted polynomial family. It establishes exact vanishing for integer powers via a differencing operator leading to when (and demonstrates this explicitly for with ), and then extends the result to noninteger using Newton's generalized binomial theorem, showing the signed sums decay as as and thus approach zero. The main contributions are the constructive Thue–Morse framework for integer powers, the noninteger generalization (Theorem thm3), and the asymptotic decay guarantee that enables near-zero signed sums for large bases. This advances the scope of PTE-type identities beyond integers and provides a method to approximate equalities for noninteger exponents in partition problems.

Abstract

We present an extension of the Prouhet-Tarry-Escott problem by demonstrating that signed sums of noninteger powers of consecutive integers can be made arbitrarily close to zero.

Paper Structure

This paper contains 3 sections, 3 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-negative Integer powers
  3. Non-integer powers

Key Result

Theorem 1

The degree of $f^j_n$ is $j-n$. In particular, $f^j_j$ is a non-zero constant polynomial, and $f^j_{n}$ is the zero polynomial provided $j<n$.

Figures (1)

  • Figure 1: $f^3_j(x)\to 0$ as $x\to\infty$

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2: Newton's generalized binomial theorem
  • Theorem 3
  • proof