The Tarry-Escott Problem

Dorwart, Harold L.; Brown, O. E.

doi:10.2307/2301480

Cited by 18 publications

(13 citation statements)

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“…, j such that Proof. When j = 2, this is a simple consequence of the binomial theorem and is a well-known lemma [7]. When j > 2, then also, the lemma follows immediately from the binomial theorem.…”

Section: Introductionmentioning

confidence: 67%

An improvement of Prouhet’s 1851 result on multigrade chains

Choudhry

2020

Int. J. Number Theory

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In 1851 Prouhet showed that when N = j k+1 where j and k are positive integers, j ≥ 2, the first N consecutive positive integers can be separated into j sets, each set containing j k integers, such that the sum of the r-th powers of the members of each set is the same for r = 1, 2, . . . , k. In this paper we show that even when N has the much smaller value 2j k , the first N consecutive positive integers can be separated into j sets, each set containing 2j k−1 integers, such that the integers of each set have equal sums of r-th powers for r = 1, 2, . . . , k. Moreover, we show that this can be done in at least {(j − 1)!} k−1 ways. We also show that there are infinitely many other positive integers N = js such that the first N consecutive positive integers can similarly be separated into j sets of integers, each set containing s integers, with equal sums of r-th powers for r = 1, 2, . . . , k, with the value of k depending on the integer N .

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Section: Introductionmentioning

confidence: 67%

An improvement of Prouhet’s 1851 result on multigrade chains

Choudhry

2020

Int. J. Number Theory

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“…We always assume that m is positive, so that the two subsets have the same size. The equations A m = B are an example of a "multigrade identity" and also an instance of the general Prouhet-Tarry-Escott problem [6,3].…”

Section: Notation and Definitionsmentioning

confidence: 99%

Littlewood Polynomials, Spectral-Null Codes, and Equipowerful Partitions

Buhler¹,

Shahar²,

Pratt³

et al. 2019

Preprint

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Let [n] denote {0, 1, ..., n−1}. A polynomial f (x) = a i x i is a Littlewood polynomial (LP) of length n if the a i are ±1 for i ∈ [n], and a i = 0 for i ≥ n. Such an LP is said to have order m if it is divisible by (x − 1) m . The problem of finding the set Lm of lengths of LPs of order m is equivalent to finding the lengths of spectral-null codes of order m, and to finding n such that [n] admits a partition into two subsets whose first m moments are equal. Extending the techniques and results of Boyd and others, we completely determine L 7 and L 8 and prove that 192 is the smallest element of L 9 . Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific n and m), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call "regenerative pairs," which produce infinite arithmetic progressions in Lm. We prove that for m ≤ 8, whenever there is an LP of length n and order m, there is one of length n and order m that is symmetric (resp. antisymmetric) if m is even (resp. odd).

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“…Similar solutions and generalizations of the Prouhet-Tarry-Escot problem are considered in[2],[3],[5],[6],[8] …”

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confidence: 99%

Combinatorial etude

Stoenchev¹,

Todorov²

2021

Annals of Computer Science and Information Systems

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The purpose of this article is to consider a special class of combinatorial problems, the so called Prouhet-Tarry-Escot problem, solution of which is realized by constructing finite sequences of ±1. For example, for fixed p ∈ N, is well known the existence of np ∈ N with the property: any set of np consecutive integers can be divided into 2 sets, with equal sums of its p thpowers. The considered property remains valid also for sets of finite arithmetic progressions of complex numbers.

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The Tarry-Escott Problem

Cited by 18 publications

References 0 publications

An improvement of Prouhet’s 1851 result on multigrade chains

An improvement of Prouhet’s 1851 result on multigrade chains

Littlewood Polynomials, Spectral-Null Codes, and Equipowerful Partitions

Combinatorial etude

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