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

TL;DR

This paper investigates the divisibility properties of the rank of appearance of primes in Lucas sequences, providing an asymptotic formula for the distribution of primes with certain divisibility conditions.

Contribution

It establishes an asymptotic formula for counting primes where a fixed odd integer divides the rank of appearance in Lucas sequences, under mild hypotheses.

Findings

Asymptotic formula for prime counts with divisibility conditions

Results applicable to Lucas sequences under mild hypotheses

Enhanced understanding of prime distribution related to Lucas sequences

Abstract

Let $U = (U_n)_{n \geq 0}$ be a Lucas sequence and, for every prime number $p$, let $\rho_U(p)$ 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 $p \leq x$ such that $d$ divides $\rho_U(p)$, as $x \to +\infty$.