On the order of finitely generated subgroups of \(\mathbb{Q}^*\pmod p\) and divisors of \(p-1\)
type: Item
instance of: scholarly article
title: On the order of finitely generated subgroups of \(\mathbb{Q}^*\pmod p\) and divisors of \(p-1\)
zbMATH Open document ID: 0847.11049
author: Francesco Pappalardi
published in: Journal of Number Theory
publication date: 1996-05-06T00:00:00Z
review text: Let \(a_1,\dots, a_r\) be multiplicatively independent non-zero integers, and let \(G_p\) be the multiplicative group that they generate modulo \(p\). It is not hard to show that if \(f(p)\) tends to infinity with \(p\), then \(\# G_p\geq (p/ f(p))^{r/(r+ 1)}\) for almost all \(p\). The paper improves this to \[ \# G_p\geq p^{r/(r+ 1)} \exp \Biggl\{{\log^\tau p\over \exp(f(p) \sqrt{\log\log p})}\Biggr\}, \] where \(\tau= (1- \log 2)/2> 0\). A number of related results are given. The key ingredient in the proofs is a delicate upper bound for the number of primes \(p\) up to \(x\) for which \(p- 1\) has a divisor in a specified range.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 874192
zbMATH Keywords: primitive roots
zbMATH Keywords: multiplicatively independent integers
zbMATH Keywords: lower bound on the order of a multiplicative group
MaRDI profile type: Publication
full work available at URL: jnth.1996.0044
OpenAlex ID: W2020346256
DOI: 10.1006/JNTH.1996.0044
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description: scientific article; zbMATH DE number 874192