Fibonacci Cycles and Fixed Points

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.