Positive lower density for prime divisors of generic linear recurrences

: Järviniemi Olli

: 2023

Mathematical Proceedings of the Cambridge Philosophical Society

: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

: MATH PROC CAMBRIDGE

: 12

: 0305-0041

: 1469-8064

DOI: https://doi.org/10.1017/S0305004123000257

: https://doi.org/10.1017/S0305004123000257

: https://research.utu.fi/converis/portal/detail/Publication/179731581

Let d = 3 be an integer and let P ? Z[x] be a polynomial of degree d whose Galois group is Sd. Let (a(n)) be a non-degenerate 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 non-zero element of the sequence (a(n)) is positive.

positive-lower-density-for-prime-divisors-of-generic-linear-recurrences.pdf