Distribution of special sequences modulo a large prime
type: Item
instance of: scholarly article
title: Distribution of special sequences modulo a large prime
zbMATH Open document ID: 1037.11002
DOI: 10.1155/S0161171203206074
author: Moubariz Z. Garaev
author: Ka-Lam Kueh
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/50748
review text: Let \(p\) be a large prime, \(g\) a primitive root (mod \(p\)), \(N>p\) a given positive integer, \[ A=\{g^x-g^y \text{ (mod \(p\)) }: 1\leq x,y\leq N \} \] and \[ B=\{xy \text{ (mod \(p\)) }: 1\leq x,y\leq N \}. \] The authors show, by elementary estimates, that \[ \# \{h \text{ (mod \(p\)) }: h\notin A \} \ll \frac{p^3}{N^3}, \tag{1} \] \[ \# \{h \text{ (mod \(p\)) }: h\notin B \} \ll \frac{p^3\log p}{N^3}. \tag{2} \] These results are in connection with the following questions: for which values of \(N\) do the sets \(A\), respectively \(B\), contain all residue classes (mod \(p\))? Estimate (1) improves a result of \textit{M. Vâjâitu} and \textit{A. Zaharescu} [Int. J. Math. Math. Sci. 29, 325--331 (2002; Zbl 1010.11003)].
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 2012610
zbMATH Keywords: residue class
zbMATH Keywords: primitive root
zbMATH Keywords: trigonometric sum
reviewed by: László Tóth
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2012610