Minimum Steps to reach to a Smaller Number in 3n+1/Collatz Process

TL;DR

This paper investigates the structure of the Collatz process, establishing bounds on stopping times and proving that only the trivial cycle at 1 exists, with no other finite cycles possible.

Contribution

It introduces a recursive binary sequence description and derives new bounds, showing the uniqueness of the trivial cycle and asymptotic behavior of the process.

Findings

Every integer congruent to 3 mod 4 arises uniquely.

Bounds imply the ratio F_q(m)/m approaches 1 as sequence length increases.

No finite nontrivial cycle exists; only the cycle at 1 is possible.

Abstract

We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for the associated stopping and cycle numbers. These bounds imply that $F_{q}(m)/m \to 1$ as the sequence length increases, while equality is impossible for any finite sequence. Consequently, no finite nontrivial cycle is compatible with the iteration, and the trivial cycle at $1$ is the only admissible periodic orbit.