The linear diophantine problem of Frobenius for subsets of arithmetic sequences
type: Item
instance of: scholarly article
title: The linear diophantine problem of Frobenius for subsets of arithmetic sequences
zbMATH Open document ID: 0879.11012
author: Stefan Matthias Ritter
published in: Archiv der Mathematik
publication date: 1998-01-18T00:00:00Z
review text: Let \(A_k= \{a_1, \dots, a_k\} \subset N\) with gcd \((a_1, \dots, a_k)=1\) and \(g(A_k)\) the corresponding Frobenius number. Now the special case \(A_k= \{a,ha+ d,ha + 2d, \dots, ha+ (k-1)d\}\) with \(h,d>0\) and gcd \((a,d) =1\) is considered, and \(\ell_k\) is defined as the greatest number of elements which can be omitted without altering \(g(A_k)\). Then it is shown that \[ 1-{4\over \sqrt k} \leq{\ell_k\over k} \leq 1-{3 \over k} \] provided \(a>k\), or \(a=k\) with \(d>2h \sqrt k\). In the case \(a> (k-4)k+3\) the lower bound can be improved to \(1-{4 \over k}\). Moreover, sets \(E_k\subset A_k\) are determined such that \(g(A_k \backslash E_k) =g(A_k)\).
Mathematics Subject Classification ID: 11D04
zbMATH DE Number: 1052893
zbMATH Keywords: linear diophantine problem of Frobenius
zbMATH Keywords: subsets of arithmetic sequences
zbMATH Keywords: Frobenius number
reviewed by: Joachim Piehler
MaRDI profile type: Publication
full work available at URL: s000130050090
OpenAlex ID: W1987086411
DOI: 10.1007/S000130050090
description: scientific article; zbMATH DE number 1052893