The $3x + 1$ problem concerns the behavior under iteration of the function $T: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ defined by $T(n) = n/2$ if $n$ is even and $T(n) = (3n + 1)/2$ if $n$ is odd. The $3x + 1$ conjecture asserts that for each $n \geq 1$ some $k$ exists with $T^{(k)}(n) = 1$; let $\sigma_\infty(n)$ equal the minimal such $k$ if one exists and $+\infty$ otherwise. The behavior of $\sigma_\infty(n)$ is irregular and seems to defy simple description. This paper describes two kinds of stochastic models that mimic some of its features. The first is a random walk that imitates the behavior of $T (\operatorname{mod}2^j)$; the second is a family of branching random walks that imitate the behavior of $T^{-1} (\operatorname{mod}3^j)$. For these models we prove analogues of the conjecture that $\lim \sup_{n \rightarrow \infty}(\sigma_\infty(n)/\log(n)) = \gamma$ for a finite constant $\gamma$. Both models produce the same constant $\gamma_0 \doteq 41.677647$. Predictions of the stochastic models agree with empirical data for the $3x + 1$ problem up to $10^{11}$. The paper also studies how many $n$ have $\sigma_\infty(n) = k$ as $k \rightarrow \infty$ and estimates how fast $t(n) = \max(T^{(k)}(n): k \geq 0)$ grows as $n \rightarrow \infty$.

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THE 3x + 1 PROBLEM: TWO STOCHASTIC MODELS

This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem The stochastic models are rigorously analyzable, and yield heuristic predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits

Stochastic Models for the 3x+1 and 5x+1 Problems

Thsi paper reviews connections of the 3x+1 mapping and its generalizations with ergodic theory and Markov chains. The work arose out of efforts to describe the observed limiting frequencies of divergent trajectories in the congruence classes (mod m).

Generalized 3x+1 mappings: Markov chains and ergodic theory

Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

1. Abstract. This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T : Z → Z, which generalizes the famous 3x + 1 mapping of Collatz. We extended T to a mapping of the polyadic numbers Ẑ and construct finitely many ergodic Borel measures on Ẑ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

Some Borel measures associated with the generalized Collatz mapping

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K. R. Matthews

Papers

A generalization of Hasse's generalization of the Syracuse algorithm