Prime divisors of the Lagarias sequence
| dc.creator | Moree, Pieter | |
| dc.creator | Stevenhagen, Peter | |
| dc.date | 1999-12-22 | |
| dc.date.accessioned | 2026-07-07T07:38:52Z | |
| dc.date.available | 2026-07-07T07:38:52Z | |
| dc.description | For integer a let us consider the sequence X_a={x_0,x_1,x_2,...} defined by x_0=a, x_1=1 and, for n>=1, x_{n+1}=x_n+x_{n-1}. We say that a prime p divides X_a if p divides at least one term of the sequence. It is easy to see that every prime p divides X_1, the sequence of Fibonacci numbers. Lagarias, using a technique involving the computation of degrees of various Kummerian extensions first employed by Hasse, showed in 1985 that X_2, the set of primes dividing some Lucas number has natural density 2/3 and posed as a challenge finding the density of prime divisors of X_3. In this paper we resolve this challenge, assuming GRH, by showing that the density of X_3 equals 1573727S/1569610, with S the so called Stephens constant. This is the first example of a `non-torsion' second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors. | |
| dc.identifier | https://arxiv.org/abs/math/9912253 | |
| dc.identifier | http://arxiv.org/abs/math/9912253 | |
| dc.identifier | J. Theor. Nombres Bordeaux 13 (2001), 241-251 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121247 | |
| dc.subject | Number Theory | |
| dc.title | Prime divisors of the Lagarias sequence | |
| dc.type | text |