Lower bounds for the total stopping time of $3x + 1$ iterates

Abstract:

The total stopping time $\sigma _{\infty }(n)$ of a positive integer $n$ is the minimal number of iterates of the $3x+1$ function needed to reach the value $1$, and is $+\infty$ if no iterate of $n$ reaches $1$. It is shown that there are infinitely many positive integers $n$ having a finite total stopping time $\sigma _{\infty }(n)$ such that $\sigma _{\infty }(n) > 6.14316 \log n.$ The proof involves a search of $3x +1$ trees to depth 60, A heuristic argument suggests that for any constant $\gamma < \gamma _{BP} \approx 41.677647$, a search of all $3x +1$ trees to sufficient depth could produce a proof that there are infinitely many $n$ such that $\sigma _{\infty }(n)>\gamma \log n.$ It would require a very large computation to search $3x + 1$ trees to a sufficient depth to produce a proof that the expected behavior of a “random” $3x +1$ iterate, which is $\gamma =\frac {2}{\log 4/3} \approx 6.95212,$ occurs infinitely often.