  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.
