Collatz conjecture failure mode

There are to ways in which the Collatz conjecture can fail:

Collatz cycle: there is a cycle that loops forever and never reaches 1
Unbounded Collatz trajectory: there is a sequence that grows without bound without looping

These are the only two options because if any sequence has an upper bound, it must sooner or later repeat an element, leading to a cycle.

Table of contents
- Collatz cycle Collatz conjecture failure mode
  - Collatz n-cycle Collatz cycle
    - Collatz 1-cycle Collatz n-cycle
    - Collatz cycle length lower bound Collatz n-cycle
- Unbounded Collatz trajectory Collatz conjecture failure mode

Collatz cycle

Collatz n-cycle

Collatz 1-cycle

Collatz cycle length lower bound

www.sciencedirect.com/science/article/pii/0012365X9390052U?via%3Dihub

Unbounded Collatz trajectory

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