2025/10/28 by Pudelko, Marc T.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2510.24882
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 m2 possible initializations in ℤ/mℤ. We discover perfect chiral symmetry between the Fibonacci recurrence an = an-1 + an-2 and its parity transform an = - an-1 + an-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.