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 * 2^{53}[approx] 2.702 * 10^{16}, the 3x+1 conjecture was verified up to that same number.