Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
type: Item
instance of: scholarly article
title: Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
zbMATH Open document ID: 1044.11005
author: Hans Roskam
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-08-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/230722
full work available at URL: jtnb13-1 english.html
review text: Let \(S\) be a linear integer recurrent sequence of order \(k\geq 2\), and define \(P_S\) as the set of primes that divide at least one term of \(S\). We say that \(S\) is degenerate if the associated characteristic polynomial is either inseparable or has two different roots whose quotient is a root of unity. Pólya proved in 1921 that the set \(P_S\) is infinite for all non-degenerate linear recurrences \(S\) of order \(k\geq 2\). The author is interested in the question of whether \(P_S\) has a natural density (Pólya's method seems inadequate to determine this). In case \(k=2\) there are many results available, cf. \textit{C. Ballot} [Density of prime divisors of linear recurrences, Mem. Am. Math. Soc. 551 (1995; Zbl 0827.11006)]. However, essentially nothing is known for sequences of order larger than two (the main focus of this paper). The author associates a free \(R\)-algebra \(O\) to \(f\) and shows that \(p\in P_S\) iff the intersection of a certain coset \(C\) of the multiplicative subgroup \((O/pO)^*\) and \(H\) is non-empty, where \(H\) denotes the kernel of the trace map from \(O/pO\) to the finite field with \(p\) elements, and is an additive subgroup of the ring \(O/pO\). The mixture of an additive and multiplicative structure is notoriously difficult to study and this forces the author into a more conjectural approach. He uses his criterion for \(p\) to be in \(P_S\) to heuristically study the divisibility of \(S\) (in case the associated characteristic polynomial is separable). Two cases are distinguished; either \(f\) splits completely into distinct linear factors modulo \(p\), or not. Denote these sets of primes by \(T_1\) and \(T_2\) respectively. The author argues that the set of primes in \(T_1\) that divide \(S\) has a positive lower density, strictly less than \(1\) and that the primes in \(T_2\) that do not divide \(S\) have zero density. Using some results from algebraic geometry the author proves that if \(S\) is a linear recurrent sequence …
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 2081367
zbMATH Keywords: recurrence sequence
zbMATH Keywords: prime divisor
zbMATH Keywords: density
zbMATH Keywords: Artin's conjecture
MaRDI profile type: Publication
OpenAlex ID: W2036863404
Wikidata QID: Q123356414
cites work: Density of prime divisors of linear recurrences
cites work: Some empirical observations on primitive roots
cites work: Q4143433
cites work: Über die Dichte der Primzahlen \(p\), für die eine vorgegebene ganzrationale Zahl \(a\neq 0\) von gerader bzw. ungerader Ordnung \(\mod p\) ist
cites work: On Artin's conjecture.
cites work: The set of primes dividing the Lucas numbers has density 2/3
cites work: On Artin's conjecture and Euclid's algorithm in global fields
cites work: Number of Points of Varieties in Finite Fields
cites work: Prime divisors of Lucas sequences
cites work: A quadratic analogue of Artin's conjecture on primitive roots.
cites work: Artin's primitive root conjecture for quadratic fields
cites work: Prime divisors of second-order linear recurrences. I
cites work: Pseudoprimes and a generalization of Artin's conjecture
cites work: Prime divisors of second order recurring sequences
cites work: The Maximal Prime Divisors of Linear Recurrences
DOI: 10.5802/JTNB.323
Recommended article: Q3338317
Recommended article: Artin's primitive root conjecture for function fields revisited
Recommended article: Artin's primitive root conjecture for quadratic fields
Recommended article: On generalizing Artins conjecture on primitive roots to composite moduli
Recommended article: A problem analogous to Artin's conjecture for primitive roots and its applications
Recommended article: Q5298140
Recommended article: An analogue of Artin's primitive root conjecture
Recommended article: Q5248561
Recommended article: An Extension of the Theorem on Primitive Divisors in Algebraic Number Fields
Recommended article: Prime power divisors of multinomial coefficients and Artin's conjecture
description: scientific article; zbMATH DE number 2081367