Modular Periodicity of Random Initialized Recurrences

TL;DR

This paper explores the complete periodic structure of Fibonacci-like recurrences with all initializations modulo m, revealing symmetry, fractal patterns, and connections to number theory and recurrence classifications.

Contribution

It introduces a comprehensive analysis of all initializations, uncovers symmetry and self-similarity, and links periodicity to quadratic reciprocity and necklace enumeration.

Findings

Discovered mirror symmetry between Fibonacci and parity transform recurrences.

Observed fractal self-similarity in periodic structures across prime powers.

Classified prime moduli based on quadratic reciprocity and established weight preservation.

Abstract

Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in $(\mathbb{Z}/m\mathbb{Z})^2$. We discover perfect mirror symmetry between the Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ and its parity transform $a_n = - a_{n-1} + a_{n-2}$ and observe fractal self-similarity in the extension from prime to prime power moduli. Additionally, we classify prime moduli based on their quadratic reciprocity and demonstrate that periodic sequences exhibit weight preservation under modular extension. Furthermore, we define a minima distribution $P(n)$ governed by Lucas ratios, which satisfies the symmetric relation $P(n)=P(1-n)$. For cyclotomic recurrences, we propose explicit counting functions for the number of distinct periods with connections to necklace enumeration. These findings imply potential connections to Viswanath's random recurrence, modular forms and L-functions.