Modular Periodicity of Random Initialized Recurrences
Marc T. Pudelko
TL;DR
The paper broadens the study of periodic sequences modulo $m$ by considering all initial conditions for restricted recurrences, separating cyclotomic and non-cyclotomic cases. It proposes explicit conjectured counting formulas for cyclotomic period landscapes, reveals a striking chiral symmetry between Fibonacci and parity recurrences, and classifies primes via Legendre symbols and an embedding parameter $\alpha$ with self-similarity across prime powers. It also introduces a weight-preservation principle under modular extension and a Lucas-ratio–driven minima distribution, and sketches potential connections to modular forms, L-functions, and higher-order recurrences. The work lays groundwork for a unified view of Pisano-like period structures and their arithmetic embeddings across moduli and polynomial families, with several open computational and theoretical questions remaining.
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 $\mathbb{Z}/m\mathbb{Z}$. We discover perfect chiral 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.