The 3x+1 problem: a lower bound hypothesis

TL;DR

This paper proposes a new hypothesis relating the initial terms of 3x+1 sequences to binary entropy, providing bounds on stopping times and excursions, aligning with prior stochastic models.

Contribution

It introduces a lower bound hypothesis based on binary entropy for the initial terms of 3x+1 sequences, offering new bounds and insights.

Findings

Supports the hypothesis with existing computations

Provides bounds for total stopping time

Aligns with previous stochastic models

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.