The $3x + 1$ Problem: Two Stochastic Models

Abstract

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 $n \rightarrow \infty$.