Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 303-314.

Let S be a linear integer recurrent sequence of order k3, 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.

Soit S une suite définie par une récurrence linéaire entière d’ordre k3. On note P S l’ensemble des nombres premiers qui divisent au moins l’un des termes de S. Nous donnons une approche heuristique du problème selon lequel P S admet ou non une densité naturelle, et montrons que certains aspects de ces heuristiques sont corrects. Sous l’hypothèse d’une certaine généralisation de la conjecture d’Artin pour les racines primitives, nous montrons que P S possède une densité asymptotique inférieure pour toute suite S “générique”. Nous donnons en illustration des exemples numériques.

@article{JTNB_2001__13_1_303_0,
     author = {Hans Roskam},
     title = {Prime divisors of linear recurrences and {Artin's} primitive root conjecture for number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {303--314},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     doi = {10.5802/jtnb.323},
     zbl = {1044.11005},
     mrnumber = {1838089},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.323/}
}
TY  - JOUR
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
DA  - 2001///
SP  - 303
EP  - 314
VL  - 13
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.323/
UR  - https://zbmath.org/?q=an%3A1044.11005
UR  - https://www.ams.org/mathscinet-getitem?mr=1838089
UR  - https://doi.org/10.5802/jtnb.323
DO  - 10.5802/jtnb.323
LA  - en
ID  - JTNB_2001__13_1_303_0
ER  - 
%0 Journal Article
%T Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
%J Journal de Théorie des Nombres de Bordeaux
%D 2001
%P 303-314
%V 13
%N 1
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.323
%R 10.5802/jtnb.323
%G en
%F JTNB_2001__13_1_303_0
Hans Roskam. Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 303-314. doi : 10.5802/jtnb.323. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.323/

[1] C. Ballot, Density of prime divisors of linear recurrent sequences. Mem. of the AMS 551 (1995). | Zbl: 0827.11006

[2] H. Brown, H. Zassenhaus, Some empirical observations on primitive roots. J. Number Theory 3 (971),306-309. | MR: 288072 | Zbl: 0219.10003

[3] R. Hartshorne, Algebraic geometry. Springer-Verlag, New York, 1977. | MR: 463157 | Zbl: 0367.14001

[4] H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganz-rationale Zahl a ≠ 0 von gerader bzw. ungerader Ordnung mod.p ist. Math. Ann. 166 (1966), 19-23. | MR: 205975 | Zbl: 0139.27501

[5] C. Hooley, On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209-220. | MR: 207630 | Zbl: 0221.10048

[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. | MR: 789184 | Zbl: 0569.10003

[7] H.W. Lenstra, JR, On Artin's conjecture and Euclid's algorithm in global fields. Inv. Math. 42 (1977), 201-224. | MR: 480413 | Zbl: 0362.12012

[8] S. Lang, A. Weil, Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819-827. | MR: 65218 | Zbl: 0058.27202

[9] P. Moree, P. Stevenhagen, Prime divisors of Lucas sequences. Acta Arith. 82 (1997), 403-410. | MR: 1483692 | Zbl: 0913.11048

[10] G. Pólya, Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen. J. Reine Angew. Math. 151 (1921), 99-100. | JFM: 47.0276.02

[11] H. Roskam, A Quadratic analogue of Artin's conjecture on primitive roots. J. Number Theory 81 (2000), 93-109. | MR: 1743503 | Zbl: 1049.11125

[12] H. Roskam, Artin's Primitive Root Conjecture for Quadratic Fields. Accepted for publication in J. Théor. Nombres Bordeaux. | Numdam | Zbl: 1026.11086

[13] P.J. Stephens, Prime divisors of second order linear recurrences. J. Number Theory 8 (1976), 313-332. | MR: 417081 | Zbl: 0334.10018

[14] S.S. Wagstaff, Pseudoprimes and a generalization of Artin's conjecture. Acta Arith. 41 (1982), 141-150. | MR: 674829 | Zbl: 0496.10001

[15] M. Ward, Prime divisors of second order recurring sequences. Duke Math. J. 21 (1954), 607-614. | MR: 64073 | Zbl: 0058.03701

[16] M. Ward, The maximal prime divisors of linear recurrences. Can. J. Math. 6 (1954), 455-462 | MR: 66408 | Zbl: 0056.04106

Cited by Sources: