Prime divisors of the Lagarias sequence
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 241-251.

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 x n n=1 ∞ définie par x 0 =3,x 1 =1 et la relation de récurrence x n+1 =x n +x n-1 . Cela donne le premier exemple d’une suite de récurrence d’ordre 2 qui n’est pas æà torsionÆ pour laquelle on sait déterminer la densité associée des diviseurs premiers.

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 x n n=1 ∞ defined by the linear recurrence x n+1 =x n +x n-1 and the initial values x 0 =3 and x 1 =1. 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.

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Moree, Pieter; Stevenhagen, Peter. Prime divisors of the Lagarias sequence. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 241-251. http://archive.numdam.org/item/JTNB_2001__13_1_241_0/

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