Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers
Brennan Benfield, Oliver Lippard
TL;DR
This work addresses how zeros in Pisano periods relate to prime factors of Fibonacci numbers by establishing that the period length factors as $\pi(m)=\alpha(m)\omega(m)$ and that the count $\omega(m)$ is determined by the congruence class of index $n$ for which primes divide $F_n$. By leveraging Renault’s classifications, Wall’s and Vinson’s results, and lcm properties of ranks and periods, the authors prove OEIS conjectures and provide a unified framework extending to $K$-Fibonacci sequences with $F_{K,0}=0$, $F_{K,1}=1$, $F_{K,n}=K F_{K,n-1}+F_{K,n-2}$, including a key residue $\beta_K(m)$ with $\omega_K(m)=\operatorname{ord}_m(\beta_K(m))$. The paper shows that, for odd $K$, the classification of $m$ by $\omega_K(m)\in\{1,2,4\}$ mirrors the classic case, with explicit criteria in terms of $\alpha_K(m)$ and $\pi_K(m)$, while even $K$ introduces nuanced behavior especially for powers of two. Final remarks extend these ideas to general $(a,b)$-Fibonacci sequences, highlighting that $\omega(m)$ divides $2\operatorname{ord}_m(-b)$ and outlining several degenerate cases, indicating rich order structures and avenues for further study.
Abstract
The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This paper proves several conjectures connecting the zeros in the Pisano period to the prime factors of $K$-Fibonacci numbers. The congruence classes of indices for $K$-Fibonacci numbers that are multiples of the prime factors of $m$ completely determine the number of zeroes in the Pisano period modulo $m$.