Generalized Modular Periodicity of Fibonacci-Type Sequences: A Comprehensive Study on Pisano Periods, Rank of Appearance, and Modular Structures

This project presents a comprehensive mathematical investigation of modular periodicity in Fibonacci-type recurrence sequences, extending classical results on the Pisano period and the rank of appearance to general integer coefficients a, b and arbitrary moduli mmm.The research unifies and expands upon existing theoretical frameworks by introducing a general modular model that captures periodic behavior across both prime and composite moduli. Building upon Wall’s foundational results (1960) and modern matrix-based modular analysis, the study rigorously derives exact and asymptotic relations between the Pisano period π(m) of appearance z(m), and the arithmetic structure of m.Through a combination of algebraic and number-theoretic reasoning, the paper formulates the Chinese Remainder Theorem decomposition of the period and establishes multiplicative lifting formulas for prime powers, including the delicate cases p = 2 and p = 5. The project introduces precise lower and upper bounds for the Pisano period in terms of the Legendre symbol (5/p), and provides a generalized formulation valid for all integers m≥2.Moreover, it systematizes the connection between the sequence’s modular recurrence and matrix dynamics over finite rings (Z/mZ)^2, bridging discrete algebra, recurrence theory, and modular arithmetic. These findings consolidate scattered theoretical insights into a unified modular framework and offer a solid foundation for future research on generalized recurrences, matrix periodicity, and algebraic dynamical systems.The project also opens new pathways for applying Fibonacci-type modular structures in cryptography, coding theory, and computational number theory, where periodicity and modular invariants play a crucial role.