Counting divisors of Lucas numbers

Abstract

Let {Sn} be a second order linear recurrence consisting of integers only. M. Ward [22] proved that, except for some degenerate cases, there are always an infinite number of distinct primes dividing the terms of {Sn}. A deeper question is whether in the non-degenerate case the set of prime divisors has a prime density. (If S is any set of natural numbers, then S(x) denotes the number of elements n in S with 1 < n ≤ x. In case S is a set of primes we define the prime density of S to be limx→∞ S(x)/π(x), if it exists, where π(x) denotes the number of primes not exceeding x.) It is conjectured that the answer is yes and that the density is in fact positive. In case of what are called torsion sequences, this was recently established by P. Stevenhagen [21], generalizing on results in the earlier papers [9, 11, 13]. Stevenhagen showed, moreover, that the density of a torsion sequence is a rational number. For a large class of non-torsion sequences, the existence and positivity of of the prime density was established by P.J. Stephens [20], under the assumption of the Generalized Riemann Hypothesis. The sequence {Ln} is torsion. Lagarias established that it has prime density 2/3. His method goes back to H. Hasse [6], who expressed the prime density of sequences {a+b}k=1 in terms of degrees of Kummer extensions. This method will be used in Section 3. The analytic aspects of prime divisors of sequences {ak + b}k=1 were explored by K. Wiertelak in several papers [24, 25, 26, 27, 28]. For a survey of results on prime divisors of, not necessarily second order, linear recurrences, see Ballot [1]. The problem of general divisors of second order linear recurrence sequences, in contrast, has not received much attention. Let a and b be fixed coprime integers such that |a| 6= |b|. In [12] the set of divisors, Ga,b, of the sequence {ak + bk} was considered. Some of the results obtained there have