## Tokyo Journal of Mathematics

### On Generalized Circuit of the Collatz Conjecture

Tomoaki MIMURO

#### 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 $f(m)-m$.

#### Article information

Source
Tokyo J. Math., Volume 28, Number 2 (2005), 593-598.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208209

Digital Object Identifier
doi:10.3836/tjm/1244208209

Mathematical Reviews number (MathSciNet)
MR2191068

Zentralblatt MATH identifier
1136.11018

#### Citation

MIMURO, Tomoaki. On Generalized Circuit of the Collatz Conjecture. Tokyo J. Math. 28 (2005), no. 2, 593--598. doi:10.3836/tjm/1244208209. https://projecteuclid.org/euclid.tjm/1244208209

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