On the prime density of Lucas sequences

Pieter Moree

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 449-459
  • ISSN: 1246-7405

Abstract

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The density of primes dividing at least one term of the Lucas sequence L n ( P ) n = 0 ∞ , defined by L 0 ( P ) = 2 , L 1 ( P ) = P and L n ( P ) = P L n - 1 ( P ) + L n - 2 ( P ) for n ≥ 2 , with P an arbitrary integer, is determined.

How to cite

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Moree, Pieter. "On the prime density of Lucas sequences." Journal de théorie des nombres de Bordeaux 8.2 (1996): 449-459. <http://eudml.org/doc/247844>.

@article{Moree1996,
abstract = {The density of primes dividing at least one term of the Lucas sequence $\left\lbrace L_n(P)\right\rbrace _\{n =0\}^\infty $, defined by $L_0(P) = 2, L_1 (P) = P$ and $L_n(P) = PL_\{n-1\}(P) + L_\{n-2\}( P)$ for $n \ge 2$, with $P$ an arbitrary integer, is determined.},
author = {Moree, Pieter},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {prime density of Lucas sequences; real quadratic field},
language = {eng},
number = {2},
pages = {449-459},
publisher = {Université Bordeaux I},
title = {On the prime density of Lucas sequences},
url = {http://eudml.org/doc/247844},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Moree, Pieter
TI - On the prime density of Lucas sequences
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 449
EP - 459
AB - The density of primes dividing at least one term of the Lucas sequence $\left\lbrace L_n(P)\right\rbrace _{n =0}^\infty $, defined by $L_0(P) = 2, L_1 (P) = P$ and $L_n(P) = PL_{n-1}(P) + L_{n-2}( P)$ for $n \ge 2$, with $P$ an arbitrary integer, is determined.
LA - eng
KW - prime density of Lucas sequences; real quadratic field
UR - http://eudml.org/doc/247844
ER -

References

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  1. [1] C. Ballot, Density of prime divisors of linear recurrences, Mem. of the Amer. Math. Soc.551, 1995. Zbl0827.11006MR1257079
  2. [2] F. Halter-Koch, Arithmetische Theorie der Normalkörper von 2-Potenzgrad mit Diedergruppe, J. Number Theory3 (1971), 412-443. Zbl0229.12006MR285511
  3. [3] H. Hasse, Uber 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] J.C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math.118 (1985), 449-461 (Errata, Pacific J. Math.162 (1994), 393-397). Zbl0569.10003MR789184
  5. [5] P. Moree, Counting divisors of Lucas numbers, MPI-preprint, no. 34, Bonn, 1996. MR1663806
  6. [6] R.W.K. Odoni, A conjecture of Krishnamurty on decimal periods and some allied problems, J. Number Theory13 (1981), 303-319. Zbl0471.10031MR634201
  7. [7] P. Ribenboim, The book of prime number records, Springer-Verlag, Berlin etc., 1988. Zbl0642.10001MR931080
  8. [8] P. Ribenboim, Catalan's conjecture, Academic Press, Boston etc., 1994. Zbl0824.11010MR1259738
  9. [9] P. Stevenhagen, The number of real quadratic fields having units of negative norm, Experimental Mathematics2 (1993), 121-136. Zbl0792.11041MR1259426
  10. [10] K. Wiertelak, On the density of some sets of primes. IV, Acta Arith.43 (1984), 177-190. Zbl0531.10049MR736730

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