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, p. 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 = {Roskam, Hans},
     title = {Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {303-314},
     zbl = {1044.11005},
     mrnumber = {1838089},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_303_0}
}
Roskam, Hans. 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. http://www.numdam.org/item/JTNB_2001__13_1_303_0/

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