Let be a linear integer recurrent sequence of order , and define as the set of primes that divide at least one term of . We give a heuristic approach to the problem whether 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 has positive lower density for “generic” sequences . Some numerical examples are included.
Soit une suite définie par une récurrence linéaire entière d’ordre . On note l’ensemble des nombres premiers qui divisent au moins l’un des termes de . Nous donnons une approche heuristique du problème selon lequel 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 possède une densité asymptotique inférieure pour toute suite “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}, pages = {303--314}, publisher = {Universit\'e Bordeaux I}, volume = {13}, number = {1}, year = {2001}, mrnumber = {1838089}, zbl = {1044.11005}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2001__13_1_303_0/} }
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 SP - 303 EP - 314 VL - 13 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_2001__13_1_303_0/ LA - en ID - JTNB_2001__13_1_303_0 ER -
%0 Journal Article %A Roskam, Hans %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 http://archive.numdam.org/item/JTNB_2001__13_1_303_0/ %G en %F 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://archive.numdam.org/item/JTNB_2001__13_1_303_0/
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