[Paper Review] On the Padovan sequence

Alain Faisant|arXiv (Cornell University)|May 19, 2019

Coding theory and cryptography1 references4 citations

TL;DR

This paper investigates the Padovan sequence $(T_n)$, defined by the recurrence $T_{n+3} = T_{n+1} + T_n$ with initial values $T_0=0$, $T_1=T_2=1$, focusing on its divisibility properties, period modulo primes, and algebraic structure via the cubic polynomial $X^3 - X - 1$. The key result establishes that the index of the first occurrence of a prime $p$ as a divisor of $T_n$, denoted $\omega_p$, satisfies $\omega_p \leq t_p \leq p^{r_p} - 1$, where $t_p$ is the period modulo $p$ and $r_p$ is the degree of the splitting field extension of $X^3 - X - 1$ over $\mathbb{F}_p$. The paper further connects these properties to class field theory and characterizes primes for which $r_p = 1$ via the condition $p = x^2 + 23y^2$.

ABSTRACT

The aim of this article is to give some properties of the so-called Padovan sequence $(T_n)_{n \ge 0}$ defined by $$ T_{n+3}=T_{n+1}+T_n \forall n \in \mathbb{N}, T_1=T_2=T_3=1$$ that is divisibility properties, periods, identities.

Motivation & Objective

• Understand the divisibility and periodic behavior of the Padovan sequence modulo primes.
• Characterize the index $\omega_p$ of the first occurrence of a prime $p$ as a divisor of $T_n$.
• Establish a connection between the splitting field degree $r_p$ of $X^3 - X - 1$ modulo $p$ and the period $t_p$ of the sequence modulo $p$.
• Use class field theory to characterize primes $p$ for which $r_p = 1$, showing $r_p = 1$ if and only if $p = x^2 + 23y^2$.
• Explore open problems related to primitive divisors, primality of $T_n$, and the finiteness of perfect powers in the sequence.

Proposed method

• The Padovan sequence is analyzed using linear recurrence theory and the theory of linearly recurrent sequences over fields.
• Roots of the characteristic polynomial $T(X) = X^3 - X - 1$ are used to express $T_n$ in closed form via symmetric functions and Vandermonde determinants.
• Modular arithmetic over $\mathbb{F}_p$ is applied to study the period $t_p$ of the sequence modulo $p$, with $r_p$ denoting the degree of the splitting field extension $R_p / \mathbb{F}_p$.
• Class field theory is employed to analyze the splitting behavior of primes in the ring of integers of $K = \mathbb{Q}(\sqrt{-23})$, linking $r_p = 1$ to the representation $p = x^2 + 23y^2$.
• Identities for $T_n$ are derived using the fact that $\alpha^n, \beta^n, \gamma^n$ lie in the 3-dimensional $K$-vector space $\mathcal{T}_K$, leading to a cubic symmetric identity in $T_n, T_{n-1}, T_{n-2}$.
• Artin reciprocity and Kummer-Dedekind theory are used to relate ideal decomposition in $\mathcal{O}_K$ to the splitting of $T(X) \mod p$, enabling the characterization of $r_p = 1$.

Experimental results

Research questions

• RQ1 Is it true that when $r_p = 1$, the coefficients $a, b, c$ in the expression $T_n = aT_{n-1}T_n + bT_{n-1}^2 + cT_{n-2}^2$ satisfy $b = c = 2a$?
• RQ2 Is it true that when $r_p = 2$, the coefficients satisfy $b = c = (p+1)a$?
• RQ3 Is it true that when $r_p = 3$, the coefficients satisfy $a = b = c = 1 + p + p^2$?
• RQ4 What is the exact arithmetic nature of $\omega_p$, the index of the first occurrence of $p$ as a divisor of $T_n$?
• RQ5 Are there finitely many $n$ for which $T_n$ has no primitive prime divisor?

Key findings

• The period $t_p$ of the Padovan sequence modulo $p$ satisfies $t_p \leq p^{r_p} - 1$, where $r_p$ is the degree of the splitting field of $X^3 - X - 1$ over $\mathbb{F}_p$.
• An explicit bound is established: $\omega_p \leq t_p \leq p^{r_p} - 1$, where $\omega_p$ is the index of the first occurrence of $p$ as a divisor of $T_n$.
• The condition $r_p = 1$ holds if and only if $p = x^2 + 23y^2$ for some integers $x, y$, as shown via class field theory and Artin reciprocity.
• The sequence contains exactly three perfect squares: $T_{10} = 9 = 3^2$, $T_{12} = 16 = 4^2$, and $T_{16} = 49 = 7^2$, with square roots equal to $T_6, T_7, T_9$ respectively.
• A symmetric cubic identity is derived: $T_n^3 + T_{n-1}^3 + T_{n-2}^3 - T_n T_{n-1}^2 - T_n^2 T_{n-2} + T_{n-1}^2 T_{n-2} + 2T_{n-1} T_{n-2}^2 - 3T_n T_{n-1} T_{n-2} = 1$.
• Primes dividing $T_n$ for $n \leq 20$ include $2, 3, 5, 7, 11, 13, 17, 19$, and the sequence exhibits a complex pattern of primitive divisors with notable exceptions such as $n = 5, 7, 10, 11, 12, 13, 14, 16, 21, 23, 32, 33, 45$.