Is the Syracuse falling time bounded by 12?

Abstract

Let $T:\mathbb{N}\to \mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a jump at $n\ge 1$ by jp$(n)=T^{(\ell )}(n)$, where $\ell$ is the number of digits of $n$ in base 2. We present computational and heuristic evidence leading to surprising conjectures. The boldest one, inspired by the study of $2^{\ell }-1$ for $\ell \le 500000$, states that for any $n\ge 2^{500}$, at most four jumps starting from $n$ are needed to fall below $n$, a strong form of the Collatz conjecture.

17 pages, 3 figures. Augmented version, with updated conjectures based on greatly extended computations