On the estimates for the number of solutions of an equation with integral coefficients
type: Item
instance of: scholarly article
title: On the estimates for the number of solutions of an equation with integral coefficients
zbMATH Open document ID: 0826.11013
DOI: 10.1007/BF01208390
author: V. V. Illarionov
published in: Mathematical Notes
publication date: 1995-09-12T00:00:00Z
review text: Let \(I_{n,k} (a_1, \dots, a_n, b)\) denote the number of solutions of \(a_1 x_1+ \cdots+ a_n x_n =b\), where \(a_i,b\in \mathbb{Z}\), \(x_i\in \{0,1, \dots, k-1\}\), \(i=1, \dots, n\) and \(J_{n,k}= \max_{a_i,b\in \mathbb{Z}} I_{n,k} (a_1, \dots, a_n, b)\). Then it is shown: (i) For \(k=\text{const.}\) and \(n\to \infty\), \(J_{n,k} \sim \sqrt {6/\pi} k^n/ \sqrt {n (k^2-1)}\). (ii) For any natural \(n\geq 1\), \(k\geq 2\), \(J_{n,k}< k^n/ \sqrt {n}\).
Mathematics Subject Classification ID: 11D04
zbMATH DE Number: 795599
zbMATH Keywords: linear diophantine equation
zbMATH Keywords: number of solutions
reviewed by: Joachim Piehler
MaRDI profile type: Publication
cites work: An Effective Formula for the Number of Solutions of Linear Boolean Equations
cites work: Q5532610
cites work: Q3240295
description: scientific article; zbMATH DE number 795599