On the least primitive root of a prime
type: Item
instance of: scholarly article
title: On the least primitive root of a prime
zbMATH Open document ID: 0079.06304
author: Harold N. Shapiro
author: P. Erdős
published in: Pacific Journal of Mathematics
publication date: 1957-01-01T00:00:00Z
review text: Let \(g(p)\) be the least positive primitive root of a prime \(p\). The authors prove that \(g(p)=O(m^c p^{1/2})\) where \(c\) is a constant and \(m\) is the number of distinct prime factors of \(p-1\). As \(m\) large, it is an improvement of a result of the reviewer: \(g(p)\leq 2^{m+1} p^{1/2}\). The authors introduce a lemma and then apply Brun's method to obtain the result. The lemma runs as following: Let \(S\) and \(T\) be two sets with distinct integers, mod \(p\). Then for any non-principal character \(\chi\), we have \[ \left|\sum_{u \in S, v \in T} \chi (u+v)\right|^2 \leq p \sum_{u \in S} 1 \sum_{v \in T} 1. \]
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 3129511
zbMATH Keywords: least primitive root of a prime
MaRDI profile type: Publication
full work available at URL: pjm.1957.7.861
OpenAlex ID: W2027550541
DOI: 10.2140/PJM.1957.7.861
description: scientific article; zbMATH DE number 3129511