On the least pair of consecutive quadratic non-residues
type: Item
instance of: scholarly article
title: On the least pair of consecutive quadratic non-residues
zbMATH Open document ID: 0617.10002
author: Adolf Hildebrand
published in: Michigan Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The author shows that for any sufficiently large prime p there is a pair of consecutive quadratic nonresidues modulo p, say n and \(n+1\), with \(n=O(p^{\theta +\epsilon})\) for \(\theta =1/4\sqrt{e}\). This improves an earlier estimate of P.D.T.A. Elliott. Like Elliot's theorem the proof uses the reviewer's character sum estimate. The author's major new tool is the following: For every \(\epsilon\in (0,1]\) there exist positive integers \(N_ 0(\epsilon)\) and \(k_ 0(\epsilon)\) with the property that for any \(N\geq N_ 0(\epsilon)\), and for any set A of positive integers satisfying \[ \sum_{n\leq N, n\equiv \ell (mod k), n\in A}1\geq \epsilon N/k\quad (1\leq k\leq k_ 0(\epsilon),\quad 0\leq \lambda \leq k-1) \] and \(\{\) n/d: \(n\in A\), d \(| n\), \(d\leq k_ 0(\epsilon)\}\subset A\), the set A contains a pair of consecutive integers \(\leq N\).
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 4000015
zbMATH Keywords: finite sets of integers
zbMATH Keywords: pair of consecutive quadratic nonresidues
zbMATH Keywords: character sum estimate
zbMATH Keywords: pair of consecutive integers
reviewed by: D. R. B. Burgess
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003482
OpenAlex ID: W2080169026
DOI: 10.1307/MMJ/1029003482
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description: scientific article; zbMATH DE number 4000015