Maximum excursion and stopping time record-holders for the problem: Computational results

Abstract

This paper presents some results concerning the search for initial values to the so-called 3 x + 1 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 n , the maximum value of the function iterates is bounded from above by n 2 f ( n ) n^2 f(n) , with f ( n ) f(n) either a constant or a very slowly increasing function of  n n . As a by-product of this (exhaustive) search, which was performed up to n = 3 ⋅ 2 53 ≈ 2.702 ⋅ 10 16 n=3 \cdot 2^{53}\approx 2.702 \cdot 10^{16} , the 3 x + 1 3x+1 conjecture was verified up to that same number.