The $3x+1$ problem: a lower bound hypothesis

Abstract

Much work has been done attempting to understand the dynamic behaviour of the so-called ``$3x+1$'' function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two. In this paper, we formulate a new hypothesis asserting that the first terms of those sequences have a lower bound which depends on the binary entropy of the ``ones-ratio''. It is in agreement with all computations so far. Furthermore it implies accurate upper bounds for the total stopping time and the maximum excursion of an integer. Theses results are consistent with two previous stochastic models of the $3x+1$ problem.