Is the Syracuse falling time bounded by 12?

TL;DR

This paper investigates the Collatz conjecture by introducing a new jump function and provides computational evidence suggesting that for large numbers, only a few steps are needed to reach smaller numbers, supporting the conjecture.

Contribution

It introduces a new accelerated jump function and presents heuristic and computational evidence supporting a strong form of the Collatz conjecture.

Findings

Computational evidence up to 500,000 digits for specific sequences.

Heuristic arguments suggest the conjecture holds for large numbers.

Proposes that at most four jumps are needed to fall below a large number.

Abstract

Let $T \colon \mathbb{N} \to \mathbb{N}$ 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.