Period Lengths of the Incomplete Fibonacci and Incomplete Lucas Sequences Modulo m

In this paper, we investigate the periodic behavior of incomplete Fibonacci and Lucas numbers modulo a positive integer m. Let L(F,k,m) and L(L,k,m) denote the least positive periods of the incomplete Fibonacci and incomplete Lucas sequences modulo m, respectively. We establish several fundamental properties of these periods, including a coprime-modulus lcm property that extends Wall’s classical result for the Fibonacci sequence. We derive explicit formulas for the least periods of both incomplete Fibonacci and incomplete Lucas sequences modulo arbitrary prime powers and arbitrary positive integers. Furthermore, we prove that the incomplete Fibonacci and incomplete Lucas sequences always have the same least period modulo m for every positive integer m>1 and every k≥0. Our approach is based on binomial coefficient representations, finite-difference operators, and periodicity properties of integer-valued polynomials. These results extend several classical periodicity properties of the Fibonacci and Lucas sequences to their incomplete counterparts and provide a systematic framework for studying their modular behavior.