On consecutive k-th power residues
type: Item
instance of: scholarly article
title: On consecutive k-th power residues
zbMATH Open document ID: 0593.10037
DOI: 10.1007/BF01490203
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178292
review text: It is known that for fixed \(k\geq 2\) and every sufficiently large prime p there exists an integer r such that r and \(r+1\) are both k-th power residues modulo p. It has been conjectured that the least positive r with this property is bounded in terms of k only. We prove this conjecture in the case k is a prime, and a weaker result in the general case. The main step in the proof consists in showing that for any completely multiplicative function f, whose values are k-th roots of unity, the equation \(f(n)=f(n+1)\), and if k is prime the stronger equation \(f(n)=f(n+1)=1\), has a solution \(n\leq c_ k\), where \(c_ k\) depends only on k. This is achieved by a combinatorial argument.
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A15
zbMATH DE Number: 3952879
zbMATH Keywords: consecutive power residues
zbMATH Keywords: completely multiplicative function
author: Adolf Hildebrand
MaRDI profile type: Publication
cites work: On the Bound for a Pair of Consecutive Quartic Residues of a Prime
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description: scientific article; zbMATH DE number 3952879