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On $d$-complete sequences modulo $l$

Volume 213 / 2024

Xing-Wang Jiang, Bing-Ling Wu Acta Arithmetica 213 (2024), 369-378 MSC: Primary 11B75 DOI: 10.4064/aa230602-31-10 Published online: 27 February 2024

Abstract

A sequence $\mathcal {T}$ of positive integers is called d-complete modulo $l$ if for every integer $0\leq u\leq l-1$, there exists an integer $v$ with $vl+u \gt 0$ such that $vl+u$ can be represented as the sum of distinct terms from $\mathcal {T}$, where no one divides any other. Recently, Chen and Yu (2023) proved that $\{m^an^b:a,b=0,1,2,\ldots \}$ is d-complete modulo $l$ if $l,m,n$ are pairwise coprime with $l,m,n\geq 2$, and posed the following problem: characterize all positive integers $l,m,n$ such that $\{m^an^b:a,b=0,1,2,\ldots \}$ is d-complete modulo $l$. We give an answer to this problem.

Authors

  • Xing-Wang JiangDepartment of Mathematics
    Luoyang Normal University
    Luoyang 471934, P.R. China
    e-mail
  • Bing-Ling WuSchool of Science
    Nanjing University of Posts and Telecommunications
    Nanjing 210023, P.R. China
    e-mail

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