A remark about the orders of \(2\pmod u\) where \(u\) is an odd number
type: Item
instance of: scholarly article
title: A remark about the orders of \(2\pmod u\) where \(u\) is an odd number
zbMATH Open document ID: 0918.11051
author: Helmut Müller
published in: Archiv der Mathematik
publication date: 1998-06-23T00:00:00Z
review text: Let \(u\) be an odd number, and let \(\ell=\ell(u)\) denote the smallest positive integer with the property \(2^{\ell}\equiv 1\bmod u\). Given a prime \(q\geqslant 3\), the author proves the following estimates concerning the asymptotic behavior of the number \(N_q(x)\) of odd, positive integers \(u\leqslant x\) with \(q\nmid \ell(u)\): \[ {x\over\log^{1/(q-1)}x}\ll N_q(x)\ll {x\over\log^{1/q}x} \qquad\text{for }x\to\infty. \] This is an essential sharpening of theorem 5 in \textit{Z. Franco} and \textit{C. Pomerance} [Math. Comput. 64, 1333-1336 (1995; Zbl 0833.11003)].
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 1145745
zbMATH Keywords: arithmetic of residue classes
zbMATH Keywords: Wieferich numbers
zbMATH Keywords: asymptotic density
zbMATH Keywords: prime number theorem for arithmetic progressions
zbMATH Keywords: asymptotic results on arithmetic functions
MaRDI profile type: Publication
full work available at URL: s000130050112
OpenAlex ID: W2094258981
DOI: 10.1007/S000130050112
description: scientific article; zbMATH DE number 1145745