Stability of two-term recurrence sequences with even parameter
type: Item
instance of: scholarly article
title: Stability of two-term recurrence sequences with even parameter
zbMATH Open document ID: 0905.11011
author: Walter Carlip
author: Eliot T. Jacobson
published in: Finite Fields and their Applications
publication date: 1999-02-04T00:00:00Z
review text: Let \(a,b\) be fixed integers and let \((u_i)^\infty_{i=0}\) be the second-order linear recurring sequence \(u_i=au_{i-1} +bu_{i-2}\) with initial values \(u_0=0\), \(u_1=1\). For any positive integer \(m\), consider the corresponding sequence \((\overline u_i)\) of residue classes \(\text{mod } m\). If \(b\) and \(m\) are relatively prime, then \((\overline u_i)\) is purely periodic. For each integer \(d\), let \(\nu(m,d)\) denote the number of occurrences of the residue \(d\pmod m\) in one (shortest) period corresponding to a fixed recurring sequence \((u_i)\) and modulus \(m\). Set \[ \Omega(m)= \{\gamma (m,d): d\in \mathbb{Z}\}. \] The sequence is called stable modulo a prime \(p\) if there is a positive integer \(N\) such that \(\Omega (p^k)= \Omega (p^N)\) for all \(k\geq N\). In this paper the stability of sequences with even parameter \(a\) is characterized. The tools of the proofs are similar to those used by the authors in [\textit{W. Carlip} and \textit{E. Jacobson}, Finite Fields Appl. 2, 369-406 (1996)], where sequences \((u_i)\) with odd parameter \(a\) have been considered.
reviewed by: Robert F. Tichy
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 1199988
zbMATH Keywords: periodic sequences
zbMATH Keywords: uniform distribution
zbMATH Keywords: second-order linear recurring sequence
zbMATH Keywords: residue classes
zbMATH Keywords: stability of sequences with even parameter
MaRDI profile type: Publication
full work available at URL: ffta.1996.0164
OpenAlex ID: W2015978197
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DOI: 10.1006/FFTA.1996.0164
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description: scientific article; zbMATH DE number 1199988