Maximum excursion and stopping time record-holders for the $3x+1$ problem: Computational results

This paper presents some results concerning the search for initial values to the so-called $3x+1$ problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of $n$, the maximum value of the function iterates is bounded from above by $n^2 f(n)$, with $f(n)$ either a constant or a very slowly increasing function of $n$. As a by-product of this (exhaustive) search, which was performed up to $n=3 \cdot 2^{53}\approx 2.702 \cdot 10^{16}$, the $3x+1$ conjecture was verified up to that same number.