There are no cycles in the $3n+1$ sequence

Abstract

In 1937, Lothar Collatz conjectured that the sequence generated by the rule $f(n)=3n+1$ for $n\in\mathbb{N}$ odd, $f(n)=n/2$ for $n\in\mathbb{N}$ even, starting in any positive integer $n$ produces $1$. This is equivalent to (1) there are no cycles except the trivial one, (1-4-2-1), and (2) there is no infinite sequence. We prove (1) using graph theory and linear algebra.