What question did this study set out to answer?

The aim is to explore integers transformed by the Collatz map that only halve without encountering odd branches.

January 23, 2026Open Access

Iii- The Pure-Halving Case of Collatz

Key Points

The aim is to explore integers transformed by the Collatz map that only halve without encountering odd branches.
Analyzed the Collatz iteration focusing on subsets of integers that can be halved exclusively.
Applied 2-adic valuation to understand properties of odd integers in the context of the Collatz map.
Derived geometric distributions related to odd integers and calculated mean contraction rate.
Examined heuristics related to random-walk dynamics to predict stopping times.
Only powers of two follow a halving pattern without encountering odd steps.
Non-powers of two eventually meet an odd number during iterations.
Identified distinct statistical behaviors in halving and random-walk scenarios that contribute to understanding Collatz dynamics.

Abstract

We analyze the subset of integers under the Collatz map: for which every iteration proceeds purely by halving. We show that only powers of two possess this property: starting from , the trajectory: contains no odd branch, and conversely every non-power of two eventually encounters an odd step. Using the 2-adic valuation , we identify a precise geometric distribution for over odd , derive the resulting mean contraction rate , and examine a probabilistic “random-walk” heuristic for expected stopping times and excursion heights. This framework separates provable structure (the halving class and valuation law) from heuristic dynamics (statistical contraction and random-walk drift), illustrating why the Collatz conjecture remains resistant to full resolution despite its strong average-case convergence.

Iii- The Pure-Halving Case of Collatz

Key Points

Abstract

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