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 defined by the linear recurrence and the initial values and . 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.
Nous donnons une solution à un problème posé par Lagarias [5] en 1985, en déterminant sous GRH la densité de l’ensemble des nombres premiers qui sont des diviseurs de termes de la suite définie par et la relation de récurrence . Cela donne le premier exemple d’une suite de récurrence d’ordre qui n’est pas æà torsionÆ pour laquelle on sait déterminer la densité associée des diviseurs premiers.
@article{JTNB_2001__13_1_241_0, author = {Moree, Pieter and Stevenhagen, Peter}, title = {Prime divisors of the {Lagarias} sequence}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {241--251}, publisher = {Universit\'e Bordeaux I}, volume = {13}, number = {1}, year = {2001}, mrnumber = {1838084}, zbl = {1064.11013}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2001__13_1_241_0/} }
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 SP - 241 EP - 251 VL - 13 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_2001__13_1_241_0/ LA - en ID - JTNB_2001__13_1_241_0 ER -
Moree, Pieter; Stevenhagen, Peter. Prime divisors of the Lagarias sequence. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 241-251. http://archive.numdam.org/item/JTNB_2001__13_1_241_0/
[1] Density of prime divisors of linear recurrent sequences. Mem. of the AMS 551, 1995. | Zbl
,[2] 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. | MR | Zbl
,[3] 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. | MR | Zbl
,[4] On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209-220. | MR | Zbl
,[5] 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 | Zbl
,[6] Algebra, 3rd edition. Addison-Wesley, 1993. | MR | Zbl
,[7] JR, On Artin's conjecture and Euclid's algorithm in global fields. Inv. Math. 42 (1977), 201-224. | MR | Zbl
,[8] Prime divisors of Lucas sequences. Acta Arith. 82 (1997), 403-410. | MR | Zbl
, ,[9] A two variable Artin conjecture. J. Number Theory 85 (2000), 291-304. | MR | Zbl
, ,[10] Prime divisors of second order linear recurrences. J. Number Theory 8 (1976), 313-332. | MR | Zbl
,[11] Prime densities for second order torsion sequences. preprint (2000).
,