Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Hans Roskam

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 303-314
  • ISSN: 1246-7405

Abstract

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Let S be a linear integer recurrent sequence of order k ≥ 3 , and define P S as the set of primes that divide at least one term of S . We give a heuristic approach to the problem whether P S has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that P S has positive lower density for “generic” sequences S . Some numerical examples are included.

How to cite

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Roskam, Hans. "Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields." Journal de théorie des nombres de Bordeaux 13.1 (2001): 303-314. <http://eudml.org/doc/248711>.

@article{Roskam2001,
abstract = {Let $S$ be a linear integer recurrent sequence of order $k \ge 3$, and define $P_S$ as the set of primes that divide at least one term of $S$. We give a heuristic approach to the problem whether $P_S$ has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that $P_S$ has positive lower density for “generic” sequences $S$. Some numerical examples are included.},
author = {Roskam, Hans},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {recurrence sequence; prime divisor; density; Artin's conjecture},
language = {eng},
number = {1},
pages = {303-314},
publisher = {Université Bordeaux I},
title = {Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields},
url = {http://eudml.org/doc/248711},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Roskam, Hans
TI - Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 303
EP - 314
AB - Let $S$ be a linear integer recurrent sequence of order $k \ge 3$, and define $P_S$ as the set of primes that divide at least one term of $S$. We give a heuristic approach to the problem whether $P_S$ has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that $P_S$ has positive lower density for “generic” sequences $S$. Some numerical examples are included.
LA - eng
KW - recurrence sequence; prime divisor; density; Artin's conjecture
UR - http://eudml.org/doc/248711
ER -

References

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  1. [1] C. Ballot, Density of prime divisors of linear recurrent sequences. Mem. of the AMS551 (1995). Zbl0827.11006
  2. [2] H. Brown, H. Zassenhaus, Some empirical observations on primitive roots. J. Number Theory3 (971),306-309. Zbl0219.10003MR288072
  3. [3] R. Hartshorne, Algebraic geometry. Springer-Verlag, New York, 1977. Zbl0367.14001MR463157
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  5. [5] C. Hooley, On Artin's conjecture. J. Reine Angew. Math.225 (1967), 209-220. Zbl0221.10048MR207630
  6. [6] J.C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3. Pacific J. Math.118 (1985), 449-461; Errata Ibid.162 (1994), 393-397. Zbl0569.10003MR789184
  7. [7] H.W. Lenstra, JR, On Artin's conjecture and Euclid's algorithm in global fields. Inv. Math.42 (1977), 201-224. Zbl0362.12012MR480413
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  9. [9] P. Moree, P. Stevenhagen, Prime divisors of Lucas sequences. Acta Arith.82 (1997), 403-410. Zbl0913.11048MR1483692
  10. [10] G. Pólya, Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen. J. Reine Angew. Math.151 (1921), 99-100. JFM47.0276.02
  11. [11] H. Roskam, A Quadratic analogue of Artin's conjecture on primitive roots. J. Number Theory81 (2000), 93-109. Zbl1049.11125MR1743503
  12. [12] H. Roskam, Artin's Primitive Root Conjecture for Quadratic Fields. Accepted for publication in J. Théor. Nombres Bordeaux. Zbl1026.11086
  13. [13] P.J. Stephens, Prime divisors of second order linear recurrences. J. Number Theory8 (1976), 313-332. Zbl0334.10018MR417081
  14. [14] S.S. Wagstaff, Pseudoprimes and a generalization of Artin's conjecture. Acta Arith.41 (1982), 141-150. Zbl0496.10001MR674829
  15. [15] M. Ward, Prime divisors of second order recurring sequences. Duke Math. J.21 (1954), 607-614. Zbl0058.03701MR64073
  16. [16] M. Ward, The maximal prime divisors of linear recurrences. Can. J. Math.6 (1954), 455-462 Zbl0056.04106MR66408

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