TY - JOUR

T1 - A generalization of the Erdös-Surányi problem

AU - Miyanohara, Eiji

PY - 2016/7/1

Y1 - 2016/7/1

N2 - Erdös-Surányi and Prielipp suggested to study the following problem: For any integers . k>0 and . n, are there an integer . N and a map . ε(lunate):(1,...,N)→(-1,1) such that . (0.1)n=∑j=1Nε(lunate)(j)jk? Mitek and Bleicher independently solved this problem affirmatively.In this paper we consider the case that for some positive odd integer . L the numbers . ε(lunate)(j) are . L-th roots of unity. We show that the answer to the corresponding question is negative if and only if . L is a prime power.

AB - Erdös-Surányi and Prielipp suggested to study the following problem: For any integers . k>0 and . n, are there an integer . N and a map . ε(lunate):(1,...,N)→(-1,1) such that . (0.1)n=∑j=1Nε(lunate)(j)jk? Mitek and Bleicher independently solved this problem affirmatively.In this paper we consider the case that for some positive odd integer . L the numbers . ε(lunate)(j) are . L-th roots of unity. We show that the answer to the corresponding question is negative if and only if . L is a prime power.

KW - Prielipp's problem

KW - Representation of integers

KW - Signed sums

KW - The set of roots of unity

KW - Theorem of Erdös and Surányi

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U2 - 10.1016/j.indag.2016.08.002

DO - 10.1016/j.indag.2016.08.002

M3 - Article

AN - SCOPUS:84996528019

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

ER -