Positive lower density for prime divisors of generic linear recurrences
Abstract
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a nondegenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one nonzero element of the sequence $(a_n)$ is positive.
 Publication:

Mathematical Proceedings of the Cambridge Philosophical Society
 Pub Date:
 November 2023
 DOI:
 10.1017/S0305004123000257
 arXiv:
 arXiv:2102.04042
 Bibcode:
 2023MPCPS.175..467J
 Keywords:

 Mathematics  Number Theory
 EPrint:
 11 pages