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The index and its prime divisors

Maciej P. Wojtkowski

TL;DR

This work reinterprets the index of appearance for second-order linear recurrences through a precise link between Chebyshev polynomials and Lucas sequences, introducing a matrix- and group-theoretic framework based on special sequence groups and DeMoivre polynomials. By employing Frobenius–Chebotarev density together with a detailed analysis of Galois groups, it derives explicit prime-density formulas for the distribution of primes p with a given r-adic valuation of the index χ(t,p), showing |Π_j(t,r)| = 1/((r+1) r^{j-1}) for r-generic t and j≥1, while thoroughly characterizing non-generic cases (including r=2 with twin, circular, and cubic symmetries). The results extend classical work of Lagarias and Ballot on finite-order sequences and establish positive-density non-divisors for broad classes of second-order recurrences, with notable applications to arithmetic dynamics such as rational orbits under rotations and Chebyshev dynamics. Overall, the paper provides a comprehensive, density-based understanding of prime divisors of second-order linear recurrences and their dynamical applications.

Abstract

We propose a new interpretation of the classical index of appearance for second order linear recursive sequences. It stems from the formula \[ C_{n}(t)-2 =\fracΔ{Q^{n}}\ L_n^2,\ \ \ \text{where} \ \ t= (T^2-2Q)/Q, \ Δ= T^2-4Q, \] 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 $Π_j(t,r), j=1,2,\dots$, 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.

The index and its prime divisors

TL;DR

This work reinterprets the index of appearance for second-order linear recurrences through a precise link between Chebyshev polynomials and Lucas sequences, introducing a matrix- and group-theoretic framework based on special sequence groups and DeMoivre polynomials. By employing Frobenius–Chebotarev density together with a detailed analysis of Galois groups, it derives explicit prime-density formulas for the distribution of primes p with a given r-adic valuation of the index χ(t,p), showing |Π_j(t,r)| = 1/((r+1) r^{j-1}) for r-generic t and j≥1, while thoroughly characterizing non-generic cases (including r=2 with twin, circular, and cubic symmetries). The results extend classical work of Lagarias and Ballot on finite-order sequences and establish positive-density non-divisors for broad classes of second-order recurrences, with notable applications to arithmetic dynamics such as rational orbits under rotations and Chebyshev dynamics. Overall, the paper provides a comprehensive, density-based understanding of prime divisors of second-order linear recurrences and their dynamical applications.

Abstract

We propose a new interpretation of the classical index of appearance for second order linear recursive sequences. It stems from the formula connecting the Chebyshev polynomials of the first kind with the Lucas sequence defined for integer by the recursion . We build on the results of \cite{L-W}. We prove that for any prime the sets , of primes such that is the highest power of dividing the index of appearance, have prime density equal to , for -generic values of . 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.

Paper Structure

This paper contains 12 sections, 35 theorems, 70 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proofs of Theorems 1 and 2 (part one)
  3. Description of the sets $\mathcal{M}_n\cap \mathcal{K}_{j}$
  4. The Galois groups of $f\Phi_{j}g_n$
  5. Non-generic values of $t$
  6. Cubic case and cubic primitivity
  7. $r=2$
  8. $r=2$, non-generic cases
  9. $r=2$, circular case
  10. Miscellanea
  11. Applications to arithmetic dynamics: the sets of divisors of iterates of linear and quadratic functions
  12. The sets of non-divisors

Key Result

Theorem 1

For $p \in \Pi(t)$ the group $\mathcal{S}_p$ is cyclic and its order is equal to $p-1$ if $\delta \mod p$ is a square in $\mathbb F_p$, and it is equal to $p+1$ if $\delta \mod p$ is not a square in $\mathbb F_p$. If $t=2 \mod p$ then the order of $\mathcal{S}_p$ is equal to $p$ and if $t=-2 \mod p$

Theorems & Definitions (68)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 58 more