Prime divisors of the Lagarias sequence

Details Prime divisors of the Lagarias sequence Prime divisors of the Lagarias sequence

1999-12-22

arxiv.org/abs/math/9912253v1

J. Theor. Nombres Bordeaux 13 (2001), 241-251

Mathematics

Number Theory

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Prime divisors of the Lagarias sequence does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Prime divisors of the Lagarias sequence, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Prime divisors of the Lagarias sequence will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-519348