Prime divisors of the Lagarias sequence

Pieter Moree; Peter Stevenhagen

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 241-251
  • ISSN: 1246-7405

Abstract

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We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence x n n = 1 ∞ defined by the linear recurrence x n + 1 = x n + x n - 1 and the initial values x 0 = 3 and x 1 = 1 . This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.

How to cite

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Moree, Pieter, and Stevenhagen, Peter. "Prime divisors of the Lagarias sequence." Journal de théorie des nombres de Bordeaux 13.1 (2001): 241-251. <http://eudml.org/doc/248699>.

@article{Moree2001,
abstract = {We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence $\left\lbrace x_n\right\rbrace ^\{\infty \}_\{n=1\}$ defined by the linear recurrence $x_\{n +1\} = x_n + x_\{n-1\}$ and the initial values $x_0 = 3$ and $x_1 = 1$. This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.},
author = {Moree, Pieter, Stevenhagen, Peter},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Chebotarev density theorem; generalized Riemann hypothesis},
language = {eng},
number = {1},
pages = {241-251},
publisher = {Université Bordeaux I},
title = {Prime divisors of the Lagarias sequence},
url = {http://eudml.org/doc/248699},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Moree, Pieter
AU - Stevenhagen, Peter
TI - Prime divisors of the Lagarias sequence
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 241
EP - 251
AB - We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence $\left\lbrace x_n\right\rbrace ^{\infty }_{n=1}$ defined by the linear recurrence $x_{n +1} = x_n + x_{n-1}$ and the initial values $x_0 = 3$ and $x_1 = 1$. This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.
LA - eng
KW - Chebotarev density theorem; generalized Riemann hypothesis
UR - http://eudml.org/doc/248699
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. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene rationale Zahl a ≠ 0 von durch eine vorgegebene Primzahl l ≠ 2 teilbarer bzw. unteilbarer Ordnung mod p ist. Math. Ann.162 (1965), 74-76. Zbl0135.10203MR186653
  3. [3] H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a ≠ 0 von gerader bzw. ungerader Ordnung mod p ist. Math. Ann.166 (1966), 19-23. Zbl0139.27501MR205975
  4. [4] C. Hooley, On Artin's conjecture. J. Reine Angew. Math.225 (1967), 209-220. Zbl0221.10048MR207630
  5. [5] 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
  6. [6] S. Lang, Algebra, 3rd edition. Addison-Wesley, 1993. Zbl0848.13001MR197234
  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
  8. [8] P. Moree, P. Stevenhagen, Prime divisors of Lucas sequences. Acta Arith.82 (1997), 403-410. Zbl0913.11048MR1483692
  9. [9] P. Moree, P. Stevenhagen, A two variable Artin conjecture. J. Number Theory85 (2000), 291-304. Zbl0966.11042MR1802718
  10. [10] P.J. Stephens, Prime divisors of second order linear recurrences. J. Number Theory8 (1976), 313-332. Zbl0334.10018MR417081
  11. [11] P. Stevenhagen, Prime densities for second order torsion sequences. preprint (2000). 

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