Fibonacci Cycles and Fixed Points

TL;DR

This paper explores cycles and fixed points of the digit-square sum function in Fibonacci bases, revealing their structure, existence conditions, and extensions to Pell polynomials, thus advancing understanding of number dynamics in special bases.

Contribution

It characterizes specific cycles and fixed points of the digit-square sum function in Fibonacci bases and extends these findings to Pell polynomial bases, providing new insights into number iteration behaviors.

Findings

Existence of cycles in Fibonacci bases related to Fibonacci numbers.

Identification of fixed points in Fibonacci bases.

Extension of cycles and fixed points to Pell polynomial bases.

Abstract

Let $S_b(n)$ denote the sum of the squares of the digits of the positive integer $n$ in base $b\geq2$. It is well-known that the sequence of iterates of $S_b(n)$ terminates in a fixed point or enters a cycle. Let $N=2n-1$, $n\geq2$. It is shown that if $b=F_{N+1}$, then a cycle of $S_b$ exists with initial term $F_{N}=F_{0}.F_{N}$, and terminal element $F_{n}.F_{n-1}$ if $n$ is even, or terminal element $F_{n-1}.F_{n}$ if $n$ is odd. Similarly, Let $N=2n+1$, $n\geq1$. If $b=F_{N-1}$, then a cycle of $S_b$ exists with initial term $F_{N}=F_{2}.F_{N-2}$, and terminal element $F_{n}.F_{n+1}$ if $n$ is even, or terminal element $F_{n+1}.F_{n}$ if $n$ is odd. Furthermore, the cycles also admit extension as an arithmetic sequence of cycles of $S_b$ with base $b=F_{N+1}+F_{N+2}k$ and $b=F_{N-1}+F_{N-2}k$, respectively. Some fixed points of $S_b$ with $b$ a Fibonacci base are shown to exist. Lastly, both cycles and fixed points admit further generalization to Pell polynomials.