Problem 1135

Erdős Problem #1135: declared status 'open'. Formalized: yes. Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd. Given any integer $m\geq 1$ does there exist $k\geq 1$ such that $f^{(k)}(m)=1$? Prize: $500. OEIS: A006370, A008908. Tags: number theory.