Fixed points of K-Fibonacci sequences

Abstract

A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$, $F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be periodic modulo every positive integer greater than $1$. If the length of one shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal to the modulus, then that positive integer is called a $\text{fixed point}$. This paper determines the fixed points of $K$-Fibonacci sequences according to the factorization of $K^2+4$ and concludes that if this process is iterated, then every modulus greater than $3$ eventually terminates at a fixed point.