The index and its prime divisors

TL;DR

This paper introduces a new interpretation of the index of appearance for second order linear recursive sequences, analyzing prime divisors and their densities, and extends previous results with applications in arithmetic dynamics.

Contribution

It provides a novel interpretation of the index of appearance, determines prime densities for prime divisors, and extends prior work on sequences of finite order with applications in arithmetic dynamics.

Findings

Prime density of sets of primes dividing the index of appearance is (r+1)r^{j-1} for r-generic t.

Complete enumeration and density formulas for non-generic cases.

The set of primes not dividing any element has positive density for second order linear recursive sequences.

Abstract

We propose a new interpretation of the classical index of appearance for second order linear recursive sequences. It stems from the formula \[ C_{n}(t)-2 =\frac{\Delta}{Q^{n}}\ L_n^2,\ \ \ \text{where} \ \ t= (T^2-2Q)/Q, \ \Delta = T^2-4Q, \] connecting the Chebyshev polynomials of the first kind $C_n(x)$ with the Lucas sequence defined for integer $T,Q\neq 0$ by the recursion $L_{n+1}= TL_n-QL_{n-1}, L_0=0, L_1 = 1$. We build on the results of \cite{L-W}. We prove that for any prime $r\geq 2$ the sets $\Pi_j(t,r), j=1,2,\dots$, of primes $p$ such that $j$ is the highest power of $r$ dividing the index of appearance, have prime density equal to $\frac{1}{(r+1)r^{j-1}}$, for $r$-generic values of $t$. We give also complete enumeration of non-generic cases and the appropriate density formulas. It improves on the work of Lagarias, \cite{L}, and Ballot, \cite{B1},\cite{B2},\cite{B3}, on the sets of prime divisors of sequences of "finite order". Our methods are sufficient to prove that for any linear recursive sequence of second order (with some trivial exceptions) the set of primes not dividing any element contains a subset of positive density. We consider also some applications in arithmetic dynamics.