On the divisibility of the rank of appearance of a Lucas sequence

2020/08/28 by Sanna, Carlo
#11B39 (Primary) 11N05 #11N37 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2008.12506

Abstract

Let U = (U_n)_{n ≥ 0} be a Lucas sequence and, for every prime number p, let ρ_U(p) be the rank of appearance of p in U, that is, the smallest positive integer k such that p divides U_k, whenever it exists. Furthermore, let d be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes p ≤ x such that d divides ρ_U(p), as x → +∞.

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