On $d$-complete sequences modulo $l$
Volume 213 / 2024
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.
