Prime divisors of the Lagarias sequence
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 241-251.

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 ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ 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.

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 ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ 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.

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Pieter Moree; Peter Stevenhagen. Prime divisors of the Lagarias sequence. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 241-251. doi : 10.5802/jtnb.318. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.318/

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