On Generalized Circuit of the Collatz Conjecture

Abstract

The Collatz conjecture is that there exists a positive integer $n$ which satisfies $f^n(m)=1$ for any integer $m \geq 3$, where $f$ is the function on the rational number field defined by $f(m)=m/2$ if the numerator of $m$ is even and $f(m)=(3m+1)/2$ if the numerator of $m$ is odd. Let $m$ be a rational number such that $f^n(m)=m>1$. Then we show that, if $m$ has some simple sequences, then the total number of positive integer $m$ is finite, by estimating $f(m)-m$.